Recent zbMATH articles in MSC 60https://www.zbmath.org/atom/cc/602022-05-16T20:40:13.078697ZUnknown authorWerkzeugOne hundred prisoners and a lightbulbhttps://www.zbmath.org/1483.000052022-05-16T20:40:13.078697Z"Dehaye, Paul-Olivier"https://www.zbmath.org/authors/?q=ai:dehaye.paul-olivier"Ford, Daniel"https://www.zbmath.org/authors/?q=ai:ford.daniel-j|ford.daniel-k|ford.daniel-a"Segerman, Henry"https://www.zbmath.org/authors/?q=ai:segerman.henrySummary: This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on.Mathematical analysis in interdisciplinary researchhttps://www.zbmath.org/1483.000422022-05-16T20:40:13.078697ZPublisher's description: This contributed volume provides an extensive account of research and expository papers in a broad domain of mathematical analysis and its various applications to a multitude of fields. Presenting the state-of-the-art knowledge in a wide range of topics, the book will be useful to graduate students and researchers in theoretical and applicable interdisciplinary research. The focus is on several subjects including: optimal control problems, optimal maintenance of communication networks, optimal emergency evacuation with uncertainty, cooperative and noncooperative partial differential systems, variational inequalities and general equilibrium models, anisotropic elasticity and harmonic functions, nonlinear stochastic differential equations, operator equations, max-product operators of Kantorovich type, perturbations of operators, integral operators, dynamical systems involving maximal monotone operators, the three-body problem, deceptive systems, hyperbolic equations, strongly generalized preinvex functions, Dirichlet characters, probability distribution functions, applied statistics, integral inequalities, generalized convexity, global hyperbolicity of spacetimes, Douglas-Rachford methods, fixed point problems, the general Rodrigues problem, Banach algebras, affine group, Gibbs semigroup, relator spaces, sparse data representation, Meier-Keeler sequential contractions, hybrid contractions, and polynomial equations. Some of the works published within this volume provide as well guidelines for further research and proposals for new directions and open problems.
The articles of mathematical interest will be reviewed individually.Introduction to the special issue on learning, optimization, and theory of G-networkshttps://www.zbmath.org/1483.000432022-05-16T20:40:13.078697ZFrom the text: Thus, both the theoretical developments and the potential for new applications have played an important role in the development of G-Networks. This special issue includes papers which represent both of these trends toward more theory and also broader applications.A tacit assumption behind Lewis triviality that is not applicable to product space conditional event algebrahttps://www.zbmath.org/1483.030352022-05-16T20:40:13.078697Z"Goodman, I. R."https://www.zbmath.org/authors/?q=ai:goodman.irwin-r"Bamber, Donald"https://www.zbmath.org/authors/?q=ai:bamber.donaldThe aim of this paper is to address the unwarranted tacit conclusion of Lewis that consistent nontrivial Boolean conditional event algebras do not exist. This is accomplished by showing: (i) Lewis' basic forcing hypothesis yielding of the triviality result need not be satisfied by a fully consistent and nontrivial Boolean conditional event algebra and (ii) a key step in Lewis' proof when extended in a natural way to a more general setting where the forcing hypothesis is not imposed also need not hold in general; however when it does, it has the same effect as Lewis' original forcing hypothesis: triviality. This property-or lack thereof-is also seen to tie in directly with the equally well-known potential property of import-export in AI and conditional logic. In the sections of this paper, the authors provide the rigorous setting for the relevant concepts discussed above and present the Boolean equivalent form of Lewis' forcing hypothesis and triviality result and introduce the lifting property and related concepts. Continuing this, they develop connections between lifting property concepts and Lewis triviality or lack thereof for BCEA (Boolean conditional event algebra) in general and PSCEA (product space conditional event algebra) in particular. The conclusion is that in general the lifting property while holding for Lewis, because of his forcing hypothesis, no longer holds in general for a wide class of BCEAs including PSCEA.
For the entire collection see [Zbl 1448.62015].
Reviewer: Ioan Tomescu (Bucureşti)Almost square permutations are typically squarehttps://www.zbmath.org/1483.050022022-05-16T20:40:13.078697Z"Borga, Jacopo"https://www.zbmath.org/authors/?q=ai:borga.jacopo"Duchi, Enrica"https://www.zbmath.org/authors/?q=ai:duchi.enrica"Slivken, Erik"https://www.zbmath.org/authors/?q=ai:slivken.erikSummary: A record in a permutation is a maximum or a minimum, from the left or from the right. The entries of a permutation can be partitioned into two types: the ones that are records are called external points, the others are called internal points. Permutations without internal points have been studied under the name of square permutations. Here, we explore permutations with a fixed number of internals points, called almost square permutations. Unlike with square permutations, a precise enumeration for the total number of almost square permutations of size \(n+k\) with exactly \(k\) internal points is not known. However, using a probabilistic approach, we are able to determine the asymptotic enumeration. This allows us to describe the permuton limit of almost square permutations with \(k\) internal points, both when \(k\) is fixed and when \(k\) tends to infinity along a negligible sequence with respect to the size of the permutation. Finally, we show that our techniques are quite general by studying the set of \(321\)-avoiding permutations of size \(n\) with exactly \(k\) additional internal points \((k\) fixed). In this case we obtain an interesting asymptotic enumeration in terms of the Brownian excursion area. As a consequence, we show that the points of a uniform permutation in this set concentrate on the diagonal and the fluctuations of these points converge in distribution to a biased Brownian excursion.A look at generalized perfect shuffleshttps://www.zbmath.org/1483.050042022-05-16T20:40:13.078697Z"Johnson, Samuel"https://www.zbmath.org/authors/?q=ai:johnson.samuel-b|johnson.samuel-g-b|johnson.samuel-d"Manny, Lakshman"https://www.zbmath.org/authors/?q=ai:manny.lakshman"Van Cott, Cornelia A."https://www.zbmath.org/authors/?q=ai:van-cott.cornelia-a"Zhang, Qiyu"https://www.zbmath.org/authors/?q=ai:zhang.qiyuSummary: Standard perfect shuffles involve splitting a deck of \(2n\) cards into two stacks and interlacing the cards from the stacks. There are two ways that this interlacing can be done, commonly referred to as an in shuffle and an out shuffle, respectively. \textit{P. Diaconis} et al. [Adv. Appl. Math. 4, 175--196 (1983; Zbl 0521.05005)] determined the permutation group generated by in and out shuffles on a deck of \(2n\) cards for all \(n\). Diaconis et al. concluded their work by asking whether similar results hold for so-called generalized perfect shuffles. For these shuffles, we split a deck of \(mn\) cards into \(m\) stacks and similarly interlace the cards with an in \(m\)-shuffle or out \(m\)-shuffle, respectively. In this paper, we find the structure of the group generated by these two shuffles for a deck of \(m^k\) cards, together with \(m^y\)-shuffles, for all possible values of \(m\), \(k\), and \(y\). The group structure is completely determined by \(k/\operatorname{gcd}(y,k)\) and the parity of \(y/\operatorname{gcd}(y,k)\). In particular, the group structure is independent of the value of \(m\).Weakly protected nodes in random binary search treeshttps://www.zbmath.org/1483.050222022-05-16T20:40:13.078697Z"Nezhad, Ezzat Mohammad"https://www.zbmath.org/authors/?q=ai:nezhad.ezzat-mohammad"Javanian, Mehri"https://www.zbmath.org/authors/?q=ai:javanian.mehri"Nabiyyi, Ramin Imany"https://www.zbmath.org/authors/?q=ai:imany-nabiyyi.raminSummary: Here, we derive the exact mean and variance of the number of weakly protected nodes (the nodes that are not leaves and at least one of their children is not a leaf) in binary search trees grown from random permutations. Furthermore, by using contraction method, we prove normal limit law for a properly normalized version of this tree parameter.Shotgun assembly of Erdős-Rényi random graphshttps://www.zbmath.org/1483.051032022-05-16T20:40:13.078697Z"Gaudio, Julia"https://www.zbmath.org/authors/?q=ai:gaudio.julia"Mossel, Elchanan"https://www.zbmath.org/authors/?q=ai:mossel.elchananSummary: Graph shotgun assembly refers to the problem of reconstructing a graph from a collection of local neighborhoods. In this paper, we consider shotgun assembly of Erdős-Rényi random graphs \(G(n,p_n)\), where \(p_n=n^{-\alpha}\) for \(0< \alpha < 1\). We consider both reconstruction up to isomorphism as well as exact reconstruction (recovering the vertex labels as well as the structure). We show that given the collection of distance-\(1\) neighborhoods, \(G\) is exactly reconstructable for \(0< \alpha < \frac{1}{3}\), but not reconstructable for \(\frac{1}{2}< \alpha < 1\). Given the collection of distance-2 neighborhoods, \(G\) is exactly reconstructable for \(\alpha\in\left(0,\frac{1}{2}\right)\cup\left(\frac{1}{2},\frac{3}{5}\right)\), but not reconstructable for \(\frac{3}{4}< \alpha< 1\).Multicolour Poisson matchinghttps://www.zbmath.org/1483.051292022-05-16T20:40:13.078697Z"Amir, Gideon"https://www.zbmath.org/authors/?q=ai:amir.gideon"Angel, Omer"https://www.zbmath.org/authors/?q=ai:angel.omer"Holroyd, Alexander E."https://www.zbmath.org/authors/?q=ai:holroyd.alexander-eSummary: Consider several independent Poisson point processes on \(\mathbb{R}^d\), each with a different colour and perhaps a different intensity, and suppose we are given a set of allowed family types, each of which is a multiset of colours such as red-blue or red-red-green. We study translation-invariant schemes for partitioning the points into families of allowed types. This generalizes the \(1\)-colour and \(2\)-colour matching schemes studied previously (where the sets of allowed family types are the singletons \{red-red\} and \{red-blue\} respectively). We characterize when such a scheme exists, as well as the optimal tail behaviour of a typical family diameter. The latter has two different regimes that are analogous to the \(1\)-colour and \(2\)-colour cases, and correspond to the intensity vector lying in the interior and boundary of the existence region respectively.
We also address the effect of requiring the partition to be a deterministic function (i.e. a factor) of the points. Here we find the optimal tail behaviour in dimension \(1\). There is a further separation into two regimes, governed by algebraic properties of the allowed family types.Convergence of non-bipartite maps via symmetrization of labeled treeshttps://www.zbmath.org/1483.051382022-05-16T20:40:13.078697Z"Addario-Berry, Louigi"https://www.zbmath.org/authors/?q=ai:addario-berry.louigi"Albenque, Marie"https://www.zbmath.org/authors/?q=ai:albenque.marieSummary: Fix an odd integer \(p\geqslant 5\). Let \(M_n\) be a uniform \(p\)-angulation with \(n\) vertices, endowed with the uniform probability measure on its vertices. We prove that there exists \(C_p\in \mathbb{R}_+\) such that, after rescaling distances by \(C_p/n^{1/4}\), \(M_n\) converges in distribution for the Gromov-Hausdorff-Prokhorov topology towards the Brownian map. To prove the preceding fact, we introduce a bootstrapping principle for distributional convergence of random labelled plane trees. In particular, the latter allows to obtain an invariance principle for labeled multitype Galton-Watson trees, with only a weak assumption on the centering of label displacements.The completely delocalized region of the Erdős-Rényi graphhttps://www.zbmath.org/1483.051512022-05-16T20:40:13.078697Z"Alt, Johannes"https://www.zbmath.org/authors/?q=ai:alt.johannes"Ducatez, Raphaël"https://www.zbmath.org/authors/?q=ai:ducatez.raphael"Knowles, Antti"https://www.zbmath.org/authors/?q=ai:knowles.anttiSummary: We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on \(N\) vertices with edge probability \(\frac{d}{N}\). We determine the full region of delocalization by determining the critical values of \(\frac{d}{\log N}\) down to which delocalization persists: for \(\frac{d}{\log N}> \frac{1}{\log 4-1}\) all eigenvectors are completely delocalized, and for \(\frac{d}{\log N}> 1\) all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [\textit{J. Alt} et al., Commun. Math. Phys. 388, No. 1, 507--579 (2021; Zbl 1477.15029); ``Poisson statistics and localization at the spectral edge of sparse Erdős-Rényi graphs'', Preprint, \url{arXiv:2106.12519}] that localized eigenvectors exist in the corresponding spectral regions.A spectral signature of breaking of ensemble equivalence for constrained random graphshttps://www.zbmath.org/1483.051552022-05-16T20:40:13.078697Z"Dionigi, Pierfrancesco"https://www.zbmath.org/authors/?q=ai:dionigi.pierfrancesco"Garlaschelli, Diego"https://www.zbmath.org/authors/?q=ai:garlaschelli.diego"Den Hollander, Frank"https://www.zbmath.org/authors/?q=ai:den-hollander.frank"Mandjes, Michel"https://www.zbmath.org/authors/?q=ai:mandjes.michelSummary: For random systems subject to a constraint, the microcanonical ensemble requires the constraint to be met by every realisation (`hard constraint'), while the canonical ensemble requires the constraint to be met only on average (`soft constraint'). It is known that for random graphs subject to topological constraints breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-vanishing specific relative entropy of the two ensembles. We investigate to what extent breaking of ensemble equivalence is manifested through the largest eigenvalue of the adjacency matrix of the graph. We consider two examples of constraints in the dense regime: (1) fix the degrees of the vertices (= the degree sequence); (2) fix the sum of the degrees of the vertices (= twice the number of edges). Example (1) imposes an extensive number of local constraints and is known to lead to breaking of ensemble equivalence. Example (2) imposes a single global constraint and is known to lead to ensemble equivalence. Our working hypothesis is that breaking of ensemble equivalence corresponds to a non-vanishing difference of the expected values of the largest eigenvalue under the two ensembles. We verify that, in the limit as the size of the graph tends to infinity, the difference between the expected values of the largest eigenvalue in the two ensembles does not vanish for (1) and vanishes for (2). A key tool in our analysis is a transfer method that uses relative entropy to determine whether probabilistic estimates can be carried over from the canonical ensemble to the microcanonical ensemble, and illustrates how breaking of ensemble equivalence may prevent this from being possible.Joint large deviation principle for some empirical measures of the \(d\)-regular random graphshttps://www.zbmath.org/1483.051572022-05-16T20:40:13.078697Z"Ibrahim, U."https://www.zbmath.org/authors/?q=ai:ibrahim.umar"Lotsi, A."https://www.zbmath.org/authors/?q=ai:lotsi.anani"Doku-Amponsah, K."https://www.zbmath.org/authors/?q=ai:doku-amponsah.kwabenaSummary: In this paper, we define a \(d\)-regular random model by perfect matching of vertices or paring of vertices. For each vertex, we assign a \(q\)-state spin. From this \(d\)-regular graph model, we define the empirical co-operate measure, which enumerates the number of co-operation between a given couple of spins, and empirical spin measure, which enumerates the number of sites having a given spin on the \(d\)-regular random graph model. For these empirical measures, we obtain large deviation principle(LDP) in the weak topology.Fluctuations for the partition function of Ising models on Erdös-Rényi random graphshttps://www.zbmath.org/1483.051582022-05-16T20:40:13.078697Z"Kabluchko, Zakhar"https://www.zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Löwe, Matthias"https://www.zbmath.org/authors/?q=ai:lowe.matthias"Schubert, Kristina"https://www.zbmath.org/authors/?q=ai:schubert.kristinaSummary: We analyze Ising/Curie-Weiss models on the Erdős-Rényi graph with \(N\) vertices and edge probability \(p=p(N)\) that were introduced by \textit{A. Bovier} and \textit{V. Gayrard} [J. Stat. Phys. 72, No. 3--4, 643--664 (1993; Zbl 1100.82515)] and investigated in [\textit{Z. Kabluchko} et al., J. Stat. Phys. 177, No. 1, 78--94 (2019; Zbl 1426.82031)] and [\textit{Z. Kabluchko} et al., ``Fluctuations of the magnetization for Ising models on Erdős-Rényi random graphs -- the regimes of small \(p\) and the critical temperature'', Preprint, \url{arXiv:1911.10624}]. We prove Central Limit Theorems for the partition function of the model and -- at other decay regimes of \(p(N)\) -- for the logarithmic partition function. We find critical regimes for \(p(N)\) at which the behavior of the fluctuations of the partition function changes.Higher-order fluctuations in dense random graph modelshttps://www.zbmath.org/1483.051592022-05-16T20:40:13.078697Z"Kaur, Gursharn"https://www.zbmath.org/authors/?q=ai:kaur.gursharn"Röllin, Adrian"https://www.zbmath.org/authors/?q=ai:rollin.adrianSummary: Our main results are quantitative bounds in the multivariate normal approximation of centred subgraph counts in random graphs generated by a general graphon and independent vertex labels. We are interested in these statistics because they are key to understanding fluctuations of regular subgraph counts -- a cornerstone of dense graph limit theory. We also identify the resulting limiting Gaussian stochastic measures by means of the theory of generalised \(U\)-statistics and Gaussian Hilbert spaces, which we think is a suitable framework to describe and understand higher-order fluctuations in dense random graph models. With this article, we believe we answer the question ``What is the central limit theorem of dense graph limit theory?''. We complement the theory with some statistical applications to illustrate the use of centred subgraph counts in network modelling.The sharp \(K_4\)-percolation threshold on the Erdős-Rényi random graphhttps://www.zbmath.org/1483.051602022-05-16T20:40:13.078697Z"Kolesnik, Brett"https://www.zbmath.org/authors/?q=ai:kolesnik.brettSummary: We locate the critical threshold \(p_c \sim 1/ \sqrt{3n\log n}\) at which it becomes likely that the complete graph \(K_n\) can be obtained from the Erdős-Rényi graph \(\mathcal{G}_{n,p}\) by iteratively completing copies of \(K_4\) minus an edge. This refines work of \textit{J. Balogh} et al. [Random Struct. Algorithms 41, No. 4, 413--440 (2012; Zbl 1279.68243)] that bounds the threshold up to multiplicative constants.Penalising transmission to hubs in scale-free spatial random graphshttps://www.zbmath.org/1483.051612022-05-16T20:40:13.078697Z"Komjáthy, Júlia"https://www.zbmath.org/authors/?q=ai:komjathy.julia"Lapinskas, John"https://www.zbmath.org/authors/?q=ai:lapinskas.john"Lengler, Johannes"https://www.zbmath.org/authors/?q=ai:lengler.johannesSummary: We study the spread of information in finite and infinite inhomogeneous spatial random graphs. We assume that each edge has a transmission cost that is a product of an i.i.d. random variable \(L\) and a penalty factor: edges between vertices of expected degrees \(w_1\) and \(w_2\) are penalised by a factor of \((w_1w_2)^{\mu}\) for all \(\mu > 0\). We study this process for scale-free percolation, for (finite and infinite) Geometric Inhomogeneous Random Graphs, and for Hyperbolic Random Graphs, all with power law degree distributions with exponent \(\tau > 1\). For \(\tau < 3\), we find a threshold behaviour, depending on how fast the cumulative distribution function of \(L\) decays at zero. If it decays at most polynomially with exponent smaller than \((3-\tau)/(2\mu)\) then explosion happens, i.e., with positive probability we can reach infinitely many vertices with finite cost (for the infinite models), or reach a linear fraction of all vertices with bounded costs (for the finite models). On the other hand, if the cdf of \(L\) decays at zero at least polynomially with exponent larger than \((3-\tau)/(2\mu)\), then no explosion happens. This behaviour is arguably a better representation of information spreading processes in social networks than the case without penalising factor, in which explosion always happens unless the cdf of \(L\) is doubly exponentially flat around zero. Finally, we extend the results to other penalty functions, including arbitrary polynomials in \(w_1\) and \(w_2\). In some cases the interesting phenomenon occurs that the model changes behaviour (from explosive to conservative and vice versa) when we reverse the role of \(w_1\) and \(w_2\). Intuitively, this could corresponds to reversing the flow of information: gathering information might take much longer than sending it out.Barak-Erdős graphs and the infinite-bin modelhttps://www.zbmath.org/1483.051622022-05-16T20:40:13.078697Z"Mallein, Bastien"https://www.zbmath.org/authors/?q=ai:mallein.bastien"Ramassamy, Sanjay"https://www.zbmath.org/authors/?q=ai:ramassamy.sanjaySummary: A Barak-Erdős graph is a directed acyclic version of the Erdős-Rényi random graph. It is obtained by performing independent bond percolation with parameter \(p\) on the complete graph with vertices \(\{1,\dots,n\}\), in which the edge between two vertices \(i< j\) is directed from \(i\) to \(j\). The length of the longest path in this graph grows linearly with the number of vertices, at rate \(C(p)\). In this article, we use a coupling between Barak-Erdős graphs and infinite-bin models to provide explicit estimates on \(C(p)\). More precisely, we prove that the front of an infinite-bin model grows at linear speed, and that this speed can be obtained as the sum of a series. Using these results, we prove the analyticity of \(C\) for \(p> 1/2\), and compute its power series expansion. We also obtain the first two terms of the asymptotic expansion of \(C\) as \(p\to 0\), using a coupling with branching random walks with selection.Infinite stable Boltzmann planar maps are subdiffusivehttps://www.zbmath.org/1483.051642022-05-16T20:40:13.078697Z"Curien, Nicolas"https://www.zbmath.org/authors/?q=ai:curien.nicolas"Marzouk, Cyril"https://www.zbmath.org/authors/?q=ai:marzouk.cyrilSummary: The infinite discrete stable Boltzmann maps are generalisations of the well-known uniform infinite planar quadrangulation in the case where large degree faces are allowed. We show that the simple random walk on these random lattices is always subdiffusive with exponent less than \(\frac{1}{3} \). Our method is based on stationarity and geometric estimates obtained via the peeling process which are of individual interest.Uniform spanning forests on biased Euclidean latticeshttps://www.zbmath.org/1483.051652022-05-16T20:40:13.078697Z"Shi, Zhan"https://www.zbmath.org/authors/?q=ai:shi.zhan.1|shi.zhan"Sidoravicius, Vladas"https://www.zbmath.org/authors/?q=ai:sidoravicius.vladas"Song, He"https://www.zbmath.org/authors/?q=ai:song.he"Wang, Longmin"https://www.zbmath.org/authors/?q=ai:wang.longmin"Xiang, Kainan"https://www.zbmath.org/authors/?q=ai:xiang.kai-nanSummary: The uniform spanning forest measure \((\mathsf{USF})\) on a locally finite, infinite connected graph with conductance \(c\), is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding \(\mathsf{USF}\) is not necessarily concentrated on the set of spanning trees. \textit{R. Pemantle} [Ann. Probab. 19, No. 4, 1559--1574 (1991; Zbl 0758.60010)] showed that on \(\mathbb{Z}^d\), equipped with the unit conductance \(c=1,\mathsf{USF}\) is concentrated on spanning trees if and only if \(d\le 4\). In this work we study the \(\mathsf{USF}\) associated with conductances \(c(e)=\lambda^{-|e|}\), where \(|e|\) is the graph distance of the edge \(e\) from the origin, and \(\lambda\in (0,1)\) is a fixed parameter. Our main result states that in this case \(\mathsf{USF}\) consists of finitely many trees if and only if \(d=2\) or \(3\). More precisely, we prove that the uniform spanning forest has \(2^d\) trees if \(d=2\) or \(3\), and infinitely many trees if \(d\ge 4\). Our method relies on the analysis of the spectral radius and the speed of the \(\lambda\)-biased random walk on \(\mathbb{Z}^d\).A time-invariant random graph with splitting eventshttps://www.zbmath.org/1483.051672022-05-16T20:40:13.078697Z"Georgakopoulos, Agelos"https://www.zbmath.org/authors/?q=ai:georgakopoulos.agelos"Haslegrave, John"https://www.zbmath.org/authors/?q=ai:haslegrave.johnSummary: We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a Poisson random number of edges are added between them. The process is parametrised by a positive real \(\lambda\) which governs the limiting average degree. We show that for each value of \(\lambda\) there is a unique random connected rooted multigraph \(M(\lambda )\) invariant under this evolution. As a consequence, starting from any finite graph \(G\) the process will almost surely converge in distribution to \(M(\lambda )\), which does not depend on \(G\). We show that this limit has finite expected size. The same process naturally extends to one in which connectedness is not necessarily preserved, and we give a sharp threshold for connectedness of this version.
This is an asynchronous version, which is more realistic from the real-world network point of view, of a process we studied in [\textit{A. Georgakopoulos}, Lond. Math. Soc. Lect. Note Ser. 436, 190--204 (2017; Zbl 1371.05268); the authors, Comb. Probab. Comput. 29, No. 4, 587--615 (2020; Zbl 1469.60328)].Central limit theorem for the least common multiple of a uniformly sampled \(m\)-tuple of integershttps://www.zbmath.org/1483.111662022-05-16T20:40:13.078697Z"Buraczewski, Dariusz"https://www.zbmath.org/authors/?q=ai:buraczewski.dariusz"Iksanov, Alexander"https://www.zbmath.org/authors/?q=ai:iksanov.aleksander-m"Marynych, Alexander"https://www.zbmath.org/authors/?q=ai:marynych.alexander-vSummary: Let \(B_n(m)\) be a set picked uniformly at random among all \(m\)-elements subsets of \(\{1, 2, \ldots, n \}\). We provide a pathwise construction of the collection \(( B_n ( m ) )_{1 \leqslant m \leqslant n}\) and prove that the logarithm of the least common multiple of the integers in \(( B_n ( \lfloor m t \rfloor ) )_{t \geqslant 0}\), properly centered and normalized, converges to a Brownian motion when both \(m, n\) tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of \(m\) independent random variables having uniform distribution on \(\{1, 2, \ldots, n \}\). Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions.Logarithmic densities in number theory. II: Logarithmic densities of arithmetic functionshttps://www.zbmath.org/1483.112002022-05-16T20:40:13.078697Z"Daili, Noureddine"https://www.zbmath.org/authors/?q=ai:daili.noureddineSummary: In this article, we present a detailed study of the logarithmic, iterated logarithmic and derived logarithmic densities of a bounded and positive arithmetic function and obtain as particular cases results on subset \(E\) of \(\mathbb{N}^* \), and give some applications to classical number theory. Some new existence criteria are established. This mathematical concept gives a prolongation of previous results.
For Part I see [the author, ``Logarithmic Densities of subsets \(E\) of \(\mathbb N\)* '', Int. J. Math. Game Theory Algebr. 27, No. 4, 1--22 (2018)].Eigenvalue distribution of some nonlinear models of random matriceshttps://www.zbmath.org/1483.150282022-05-16T20:40:13.078697Z"Benigni, Lucas"https://www.zbmath.org/authors/?q=ai:benigni.lucas"Péché, Sandrine"https://www.zbmath.org/authors/?q=ai:peche.sandrineSummary: This paper is concerned with the asymptotic empirical eigenvalue distribution of some non linear random matrix ensemble. More precisely we consider \(M=\frac{1}{m}YY^{\ast}\) with \(Y=f(WX)\) where \(W\) and \(X\) are random rectangular matrices with i.i.d. centered entries. The function \(f\) is applied pointwise and can be seen as an activation function in (random) neural networks. We compute the asymptotic empirical distribution of this ensemble in the case where \(W\) and \(X\) have sub-Gaussian tails and \(f\) is real analytic. This extends a result of [\textit{J. Pennington} and \textit{P. Worah}, J. Stat. Mech. Theory Exp. 2019, No. 12, Article ID 124005, 14 p. (2019; Zbl 1459.60012)] where the case of Gaussian matrices \(W\) and \(X\) is considered. We also investigate the same questions in the multi-layer case, regarding neural network and machine learning applications.A quadratic identity in the shuffle algebra and an alternative proof for de Bruijn's formulahttps://www.zbmath.org/1483.170232022-05-16T20:40:13.078697Z"Colmenarejo, Laura"https://www.zbmath.org/authors/?q=ai:colmenarejo.laura"Diehl, Joscha"https://www.zbmath.org/authors/?q=ai:diehl.joscha"Sorea, Miruna-Ştefana"https://www.zbmath.org/authors/?q=ai:sorea.miruna-stefanaColmenarejo, Galuppi and Michalek proved a formula for lattice paths, involving on the left side an \(n\times n\) determinant of iterated integrals of order 2, on the other side the square of an alternated sum of iterated integrals of order \(n\).
The aim is to provide a purely algebraic proof of this identity. The natural frame for this is the shuffle algebra, as the considered integrals define a character on it. The left side is replaced by a \(n\times n\) determinant of words of length 2 (computed with the shuffle product). For the right side, a basis of the shuffle algebra is required: it comes from the action of the group \(\mathrm{SL}(\mathbb{R}^d)\) and is indexed by rectangular standard Young tableaux. The right side is then the square of a particular element of this basis, up to a power of 2.
It is finally show that this algebraic settings implies de Bruijn's formula for the Pfaffian.
Reviewer: Loïc Foissy (Calais)The absolute of finitely generated groups. II: The Laplacian and degenerate partshttps://www.zbmath.org/1483.201252022-05-16T20:40:13.078697Z"Vershik, A. M."https://www.zbmath.org/authors/?q=ai:vershik.anatoli-m"Malyutin, A. V."https://www.zbmath.org/authors/?q=ai:malyutin.andrei-valerevichSummary: The article continues a series of papers on the absolute of finitely generated groups [Eur. J. Math. 4, No. 4, 1476--1490 (2018; Zbl 1403.28014)]. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of \textit{cotransition probabilities} is the same as for the simple (right) random walk generated by the uniform distribution on the generators. The absolute is a new boundary of a group, generated by random walks on the group. We divide the absolute into two parts, Laplacian and degenerate, and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under taking certain central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelianization; consider a number of fundamental examples (free groups, commutative groups, the discrete Heisenberg group).On the growth of \(L^2\)-invariants of locally symmetric spaces. II: Exotic invariant random subgroups in rank onehttps://www.zbmath.org/1483.220072022-05-16T20:40:13.078697Z"Abert, Miklos"https://www.zbmath.org/authors/?q=ai:abert.miklos"Bergeron, Nicolas"https://www.zbmath.org/authors/?q=ai:bergeron.nicolas"Biringer, Ian"https://www.zbmath.org/authors/?q=ai:biringer.ian"Gelander, Tsachik"https://www.zbmath.org/authors/?q=ai:gelander.tsachik"Nikolov, Nikolay"https://www.zbmath.org/authors/?q=ai:nikolov.nikolay"Raimbault, Jean"https://www.zbmath.org/authors/?q=ai:raimbault.jean"Samet, Iddo"https://www.zbmath.org/authors/?q=ai:samet.iddoThis paper is a sequel to [Ann. Math. (2) 185, No. 3, 711--790 (2017; Zbl 1379.22006)] by the same authors. The former paper had extended the validity of the Lück approximation theorem to the setting of Benjamini-Schramm convergence for uniformly discrete sequences of lattices \(\Gamma_n\) in a higher-rank symmetric space of noncompact type \(X=G/K\). That is, it proved \(\lim_{n\to\infty}\frac{b_k(\Gamma_n\backslash X)}{\mathrm{Vol}(\Gamma_n\backslash X)}=\beta_k^{(2)}(X)\) under the assumption that \(\Gamma_n\backslash X\) BS-converges to \(X\).
Their approach towards this theorem had been to introduce the notion of invariant random subgroups (IRSs), that is, conjugation-invariant Borel probability measures on \(\mathrm{Sub}(G)\). (\(\mathrm{Sub}(G)\) is the space of closed subgroups with the Chabauty topology.) For a lattice \(\Gamma\subset G\) one has a map \(\Gamma\backslash G\to \mathrm{Sub}(G)\) sending \(\Gamma g\) to \(g^{-1} \Gamma g\), and one can use the finite measure on \(\Gamma\backslash G\) to define an IRS on \(\mathrm{Sub}(G)\). BS-convergence of \(\Gamma_n\backslash X\) to \(X\) is equivalent to convergence of \(\mu_{\Gamma_n}\) to \(\mu_{\mathrm{id}}\) for the weak-*-topology on \(\mathrm{IRS}(G)\). (The latter is compact, so sequences converge up to extraction.)
By ergodic decomposition it suffices to study ergodic IRSs. If \(\mathrm{rank}_{\mathbb R}(G)\ge 2\), then the Nevo-Stuck-Zimmer theorem implies that the only ergodic IRSs are \(\mu_G,\mu_{\mathrm{id}}\) and \(\mu_\Gamma\) for some lattice \(\Gamma\). Moreover, for every sequence of pairwise non-conjugate lattices, \(\mu_{\Gamma_n}\) converges to \(\mu_{\mathrm{id}}\). This was a main ingredient in the proof by the authors of the improved Lück approximation theorem.
If \(\mathrm{rank}_{\mathbb R}(G)=1\), there are much more possibilities for IRSs. First, in this case for a lattice \(\Gamma\subset G\) the Margulis normal subgroup theorem does not apply. There are many normal subgroups of infinite index which yield an IRS. Next, lattices \(\Gamma\) may have epimorphisms to the free group \(F_2\), and by the work of \textit{L. Bowen} [Groups Geom. Dyn. 9, No. 3, 891--916 (2015; Zbl 1358.37011)] there are many exotic IRSs on free groups. Using the epimorphism one obtains then exotic IRSs supported on \(\Gamma\).
The paper under review is devoted to the construction of other uncountable families of IRSs in \(\mathrm{SO}(n,1)=\mathrm{Isom}^+({\mathbb H}^n)\). It follows from the Borel density theorem, that an ergodic IRS \(\mu\not=\mu_G\) is almost-surely discrete. Thus it can be seen as a probability measure on the set of discrete subgroups or equivalently on the set of (framed) hyperbolic manifolds. The authors describe several constructions of random hyperbolic manifolds, which frequently can not be induced by lattices.
One such construction takes two hyperbolic \(n\)-manifolds \(N_0\) and \(N_1\), whose totally geodesic boundaries consist both of the same two copies of some hyperbolic \((n-1)\)-manifold. To each \(\alpha\in\left\{0,1\right\}^{\mathbb Z}\) one obtains a hyperbolic \(n\)-manifold \(N_\alpha\) by glueing copies of \(N_0\) and \(N_1\) according to the pattern prescribed by \(\alpha\). Each shift-invariant measure on \(\left\{0,1\right\}^{\mathbb Z}\) yields a random hyperbolic \(n\)-manifold. This IRS is not induced by a lattice if \(N_0\) and \(N_1\) are not embedded in non-commensurable compact arithmetic \(n\)-manifolds and \(\alpha\) is not supported on a shift-periodic orbit.
Another construction takes a topological surface \(S\) glued from infinitely many pairs of pants along the pattern of an infinite \(3\)-regular tree. Hyperbolic metrics on \(S\) are described by Fenchel-Nielsen coordinates. Choosing Fenchel-Nielsen coordinates randomly from \(\left(0,\infty\right)\times S^1\) one obtains a random hyperbolic surface. For appropriately measures on \(\left(0,\infty\right)\) and the Lebesgue measure on \(S^1\) one obtains IRSs not induced by a lattice.
A further construction takes a subgroup of the mapping class group \(\mathrm{Mod}(\Sigma)\) freely generated by pseudo-Anosov \(\phi_1,\ldots,\phi_n\), such that orbits on Teichmüller space are quasi-convex. For a sequence of words with \(\vert w_i\vert\to\infty\) let \(\Gamma_i\backslash{\mathbb H}^3\) be hyperbolic \(3\)-manifold fibering over \(S^1\) with monodromy \(w_i\). The sequence \(\mu_{\Gamma_i}\) converges (up to extraction) to an IRS \(\mu\). If the words \(w_i\) are chosen appropriately, then \(\mu\) is not induced by a lattice.
All these constructions yield weak-*-limits of sequences \(\mu_{\Gamma_n}\) for lattices \(\Gamma_n\) and the authors ask whether this must be the case for every ergodic IRS \(\mu\not=\mu_G\).
Reviewer: Thilo Kuessner (Augsburg)On a recursive construction of Dirichlet form on the Sierpiński gaskethttps://www.zbmath.org/1483.280072022-05-16T20:40:13.078697Z"Gu, Qingsong"https://www.zbmath.org/authors/?q=ai:gu.qingsong"Lau, Ka-Sing"https://www.zbmath.org/authors/?q=ai:lau.kasing"Qiu, Hua"https://www.zbmath.org/authors/?q=ai:qiu.huaSummary: Let \(\Gamma_n\) denote the \(n\)-th level Sierpiński graph of the Sierpiński gasket \(K\). We consider, for any given conductance \(a_0,b_0c_0\) on \(\Gamma_0\), the Dirichlet form \(\mathcal{E}\) on \(K\) obtained from a recursive construction of compatible sequence of conductances \((a_n, b_n, c_n)\) on \(\Gamma_n,n \geq 0\). We prove that there is a dichotomy situation: either \(a_0 = b_0 c_0\) and \(\mathcal{E}\) is the standard Dirichlet form, or \(a_0 > b_0 =c_0\) (or the two symmetric alternatives), and \(\mathcal{E}\) is a non-self-similar Dirichlet form independent of \(a_0,b_0\). The second situation has been studied in as a one-dimensional asymptotic diffusion. The analytic approach here is more direct and yields sharper results; in particular, for the spectral property, we give a precise estimate of the eigenvalue distribution of the associated Laplacian, which improves a similar result in [\textit{B. Hambly} and \textit{O. Jones}, J. Theor. Probab. 15, No. 2, 285--322 (2002; Zbl 1003.60071)].Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomialshttps://www.zbmath.org/1483.300282022-05-16T20:40:13.078697Z"Beliaev, D."https://www.zbmath.org/authors/?q=ai:beliaev.dmitri-b"Muirhead, S."https://www.zbmath.org/authors/?q=ai:muirhead.stephen"Wigman, I."https://www.zbmath.org/authors/?q=ai:wigman.igorSummary: Beginning with the predictions of Bogomolny-Schmit for the random plane wave, in recent years the deep connections between the level sets of smooth Gaussian random fields and percolation have become apparent. In classical percolation theory a key input into the analysis of global connectivity are scale-independent bounds on crossing probabilities in the critical regime, known as Russo-Seymour-Welsh (RSW) estimates. Similarly, establishing RSW-type estimates for the nodal sets of Gaussian random fields is a major step towards a rigorous understanding of these relations.
The Kostlan ensemble is an important model of Gaussian homogeneous random polynomials. The nodal set of this ensemble is a natural model for a `typical' real projective hypersurface, whose understanding can be considered as a statistical version of Hilbert's 16th problem. In this paper we establish RSW-type estimates for the nodal sets of the Kostlan ensemble in dimension two, providing a rigorous relation between random algebraic curves and percolation. The estimates are uniform with respect to the degree of the polynomials, and are valid on all relevant scales; this, in particular, resolves an open question raised recently by Beffara-Gayet. More generally, our arguments yield RSW estimates for a wide class of Gaussian ensembles of smooth random functions on the sphere or the flat torus.Conformal welding for critical Liouville quantum gravityhttps://www.zbmath.org/1483.300292022-05-16T20:40:13.078697Z"Holden, Nina"https://www.zbmath.org/authors/?q=ai:holden.nina"Powell, Ellen"https://www.zbmath.org/authors/?q=ai:powell.ellenSummary: Consider two critical Liouville quantum gravity surfaces (i.e., \(\gamma\)-LQG for \(\gamma =2)\), each with the topology of \(\mathbb{H}\) and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces, when the boundaries are identified according to quantum boundary length. This results in a critical LQG surface decorated by an independent \(\mathrm{SLE}_4\). Combined with the proof of uniqueness for such a welding, recently established by \textit{O. McEnteggart, J. Miller} and \textit{W. Qian} [``Uniqueness of the welding problem for SLE and Liouville quantum gravity'', Preprint, \url{arXiv:1809.02092}], this shows that the welding operation is well-defined. Our result is a critical analogue of \textit{S. Sheffield}'s quantum gravity zipper theorem [Ann. Probab. 44, No. 5, 3474--3545 (2016; Zbl 1388.60144)], which shows that a similar conformal welding for subcritical LQG (i.e., \(\gamma\)-LQG for \(\gamma\in (0,2))\) is well-defined.Time-reversal of multiple-force-point \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) with all force points lying on the same sidehttps://www.zbmath.org/1483.300302022-05-16T20:40:13.078697Z"Zhan, Dapeng"https://www.zbmath.org/authors/?q=ai:zhan.dapeng.1|zhan.dapengSummary: We define intermediate \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) and reversed intermediate \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) processes using Appell-Lauricella multiple hypergeometric functions, and use them to describe the time-reversal of multiple-force-point chordal \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) curves in the case that all force points are on the boundary and lie on the same side of the initial point, and \(\kappa\) and \(\underline{\rho}=(\rho_1,\dots,\rho_m)\) satisfy that either \(\kappa\in (0,4]\) and \(\sum_{{j=1}^k}{\rho_j}> -2\) for all \(1\le k\le m\), or \(\kappa\in (4,8)\) and \(\sum_{{j=1}^k}\rho_j\ge \frac{\kappa}{2}-2\) for all \(1\le k\le m\).On the Mayer series of two-dimensional Yukawa gas at inverse temperature in the interval of collapsehttps://www.zbmath.org/1483.310102022-05-16T20:40:13.078697Z"Kroschinsky, Wilhelm"https://www.zbmath.org/authors/?q=ai:kroschinsky.wilhelm"Marchetti, Domingos H. U."https://www.zbmath.org/authors/?q=ai:marchetti.domingos-h-uSummary: We prove a theorem on the minimal specific energy for a \(\pm 1\) charged particles system, interacting through a class of pair potential \(v\), that may be stated as follows: suppose \(v\) may be represented by a scale mixtures of \(d\)-dimensional Euclid's hat. If the number of particles \(n\) is even, then their interacting energy \(U_n\) divided by \(n\) is minimized by a constant \(B\) at the configurations with total charge zero and all particles collapsed to a point; if \(n\) is odd, then the ratio \(U_n/(n-1)\) is minimized by a constant \(\bar{B}=B\) at the configurations with total charge \(\pm 1\) and all particles collapsed to a point. The theorem is then used to investigate the convergence of the Mayer series for a gas of \(\pm 1\) charged particles interacting through the two-dimensional Yukawa pair potential \(v\) for inverse temperatures in the collapse interval \([4\pi ,8\pi )\). The convergence is proved in the present paper up to the second threshold \(6\pi\) using the decomposition of the Yukawa potential into scales of modified Bessel functions (standard) and into scale mixtures of Euclid's hat. Moreover, assuming that \textbf{(i)} neutral subclusters of size smaller than an odd number \(k>1\) do not collapse inside a cluster of size larger than \(k\) for \(\beta\) in the threshold interval \([8\pi (k-2)/(k-1),8\pi k/(k+1))\) and \textbf{(ii)} they satisfy a technical condition, then the Mayer series, discarding the first even coefficients of order smaller than \(k,\) converges.On the exact asymptotics of exit time from a cone of an isotropic \(\alpha\)-self-similar Markov process with a skew-product structurehttps://www.zbmath.org/1483.310172022-05-16T20:40:13.078697Z"Palmowski, Zbigniew"https://www.zbmath.org/authors/?q=ai:palmowski.zbigniew"Wang, Longmin"https://www.zbmath.org/authors/?q=ai:wang.longminSummary: In this paper we identify the asymptotic tail of the distribution of the exit time \(\tau_C\) from a cone \(C\) of an isotropic \(\alpha\)-self-similar Markov process \(X_t\) with a skew-product structure, that is, \(X_t\) is a product of its radial process and an independent time changed angular component \(\Theta_t\). Under some additional regularity assumptions, the angular process \(\Theta_t\) killed on exiting the cone \(C\) has a transition density that can be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asymptotic properties of the exponential functional of a killed Lévy process related to the Lamperti representation of the radial process, we prove that
\[
P_x (\tau_C >t)\sim h(x)t^{-\kappa_1}
\]
as \(t\rightarrow\infty\) for \(h\) and \(\kappa_1\) identified explicitly. The result extends the work of \textit{R. D. De Blassie} [``Remark on exit times from cones in \(\mathbb R^n\) of Brownian motion'', Probab. Theory Related Fields 79, 95--97 (1988)] and \textit{R. Bañuelos} and \textit{R. G. Smits} [Probab. Theory Relat. Fields 108, No. 3, 299--319 (1997; Zbl 0884.60037)] concerning the Brownian motion.Mean exit time for the overdamped Langevin process: the case with critical points on the boundaryhttps://www.zbmath.org/1483.310322022-05-16T20:40:13.078697Z"Nectoux, Boris"https://www.zbmath.org/authors/?q=ai:nectoux.borisSummary: Let \((X_t)_{t\geq 0}\) be the overdamped Langevin process on \(\mathbb{R}^d\) i.e. the solution of the stochastic differential equation
\[
dX_t=-\nabla f(X_t)\,dt+\sqrt{h}\,dB_t.
\]
Let \(\Omega\subset\mathbb{R}^d\) be a bounded domain. In this work, when \(X_0=x\in\Omega\), we derive new sharp asymptotic equivalents (with optimal error terms) in the limit \(h\to 0\) of the mean exit time from \(\Omega\) of the process \((X_t)_{t\geq 0}\) (which is the solution of \(\left(-\frac{h}{2}\Delta+\nabla f\cdot\nabla\right)w=1\) in \(\Omega\) and \(w=0\) on \(\partial\Omega\)), when the function \(f\to\Omega\to\mathbb{R}\) has critical points on \(\partial\Omega\) Such a setting is the one considered in many cases in molecular dynamics simulations. This problem has been extensively studied in the literature but such a setting has never been treated. The proof, mainly based on techniques from partial differential equations, uses recent spectral results from
[\textit{D. Le Peutrec} and the author, SIAM J. Math. Anal. 52, No. 1, 581--604 (2020; Zbl 1430.35175)] and its starting point is a formula from the potential theory. We also provide new sharp leveling results on the mean exit time from \(\Omega\).Polyharmonic functions for finite graphs and Markov chainshttps://www.zbmath.org/1483.310342022-05-16T20:40:13.078697Z"Hirschler, Thomas"https://www.zbmath.org/authors/?q=ai:hirschler.thomas"Woess, Wolfgang"https://www.zbmath.org/authors/?q=ai:woess.wolfgangSummary: On a finite graph with a chosen partition of the vertex set into interior and boundary vertices, a \(\lambda\)-polyharmonic function is a complex function \(f\) on the vertex set which satisfies \((\lambda \cdot I- P)^nf(x) = 0\) at each interior vertex. Here, \(P\) may be the normalised adjacency matrix, but more generally, we consider the transition matrix \(P\) of an arbitrary Markov chain to which the (oriented) graph structure is adapted. After describing these ``global'' polyharmonic functions, we turn to solving the \textit{Riquier} problem, where \(n\) boundary functions are preassigned and a corresponding ``tower'' of \(n\) successive Dirichlet type problems is solved. The resulting unique solution will be polyharmonic only at those points which have distance at least \(n\) from the boundary. Finally, we compare these results with those concerning infinite trees with the end boundary, as studied by \textit{J. M. Cohen} et al. [Am. J. Math. 124, No. 5, 999--1043 (2002; Zbl 1025.31003)], and more recently, by \textit{M. A. Picardello} and the second author [Potential Anal. 51, No. 4, 541--561 (2019; Zbl 1429.31007)].
For the entire collection see [Zbl 1473.53004].Discrete harmonic functions in the three-quarter planehttps://www.zbmath.org/1483.310362022-05-16T20:40:13.078697Z"Trotignon, Amélie"https://www.zbmath.org/authors/?q=ai:trotignon.amelieSummary: In this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane -- resolution of a functional equation via boundary value problem using a conformal mapping -- to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.Homogenization of symmetric Dirichlet formshttps://www.zbmath.org/1483.310372022-05-16T20:40:13.078697Z"Tomisaki, Matsuyo"https://www.zbmath.org/authors/?q=ai:tomisaki.matsuyo"Uemura, Toshihiro"https://www.zbmath.org/authors/?q=ai:uemura.toshihiroSummary: We consider a homogenization problem for symmetric jump-diffusion processes by using the Mosco convergence and the two-scale convergence of the corresponding Dirichlet forms. Moreover, we show the weak convergence of the processes.Dynamics of a vector-host model under switching environmentshttps://www.zbmath.org/1483.340692022-05-16T20:40:13.078697Z"Watts, Harrison"https://www.zbmath.org/authors/?q=ai:watts.harrison"Mishra, Arti"https://www.zbmath.org/authors/?q=ai:mishra.arti"Nguyen, Dang H."https://www.zbmath.org/authors/?q=ai:nguyen.dang-hai"Tuong, Tran D."https://www.zbmath.org/authors/?q=ai:tuong.tran-dinhSummary: In this paper, the stochastic vector-host model has been proposed and analysed using nice properties of piecewise deterministic Markov processes (PDMPs). A threshold for the stochastic model is derived whose sign determines whether the disease will eventually disappear or persist. We show mathematically the existence of scenarios where switching plays a significant role in surprisingly reversing the long-term properties of deterministic systems.Inferring the connectivity of coupled oscillators and anticipating their transition to synchrony through lag-time analysishttps://www.zbmath.org/1483.340752022-05-16T20:40:13.078697Z"Leyva, Inmaculada"https://www.zbmath.org/authors/?q=ai:leyva.inmaculada"Masoller, Cristina"https://www.zbmath.org/authors/?q=ai:masoller.cristinaSummary: The synchronization phenomenon is ubiquitous in nature. In ensembles of coupled oscillators, explosive synchronization is a particular type of transition to phase synchrony that is first-order as the coupling strength increases. Explosive sychronization has been observed in several natural systems, and recent evidence suggests that it might also occur in the brain. A natural system to study this phenomenon is the Kuramoto model that describes an ensemble of coupled phase oscillators. Here we calculate bi-variate similarity measures (the cross-correlation, \(\rho_{ij}\), and the phase locking value, \(\text{PLV}_{ij})\) between the phases, \(\varphi_i(t)\) and \(\varphi_j(t)\), of pairs of oscillators and determine the lag time between them as the time-shift, \(\tau_{ij}\), which gives maximum similarity (i.e., the maximum of \(\rho_{ij}(\tau)\) or \(\text{PLV}_{ij}(\tau))\). We find that, as the transition to synchrony is approached, changes in the distribution of lag times provide an earlier warning of the synchronization transition (either gradual or explosive). The analysis of experimental data, recorded from Rossler-like electronic chaotic oscillators, suggests that these findings are not limited to phase oscillators, as the lag times display qualitatively similar behavior with increasing coupling strength, as in the Kuramoto oscillators. We also analyze the statistical relationship between the lag times between pairs of oscillators and the existence of a direct connection between them. We find that depending on the strength of the coupling, the lags can be informative of the network connectivity.Stochastic resonance in a high-order time-delayed feedback tristable dynamic system and its applicationhttps://www.zbmath.org/1483.340792022-05-16T20:40:13.078697Z"Shi, Peiming"https://www.zbmath.org/authors/?q=ai:shi.peiming"Zhang, Wenyue"https://www.zbmath.org/authors/?q=ai:zhang.wenyue"Han, Dongying"https://www.zbmath.org/authors/?q=ai:han.dongying"Li, Mengdi"https://www.zbmath.org/authors/?q=ai:li.mengdiSummary: A stochastic resonance (SR) tristable system based on a high-order time-delayed feedback is investigated and the feasibility of the system for weak fault signature extraction is discussed. The potential function, the mean first-passage time (MFPT) and the signal-to-noise ratio (SNR) are used to evaluate the model. Firstly, the potential function and stationary probability function (PDF) of the system are derived, and then the influence of the time delay parameters on the MFPT of the particles is analyzed. Secondly, the influences of time-delyed strength \(e\) and delyed length \(\tau\) on the SR system from the perspective of the transition of the particles in the potential wells are discussed, and then the SNR and the effect of the parameters on the SNR are derived. In addition, the high-order time-delayed feedback tristable stochastic resonance (HTFTSR) system is used to deal with faulty bearing data and is compared with traditional tristable stochastic resonance (TSR). The result shows that the nonlinear system model can accurately identify the fault frequency and improve the energy of the characteristic signal under the appropriate system parameters.On the Lyapunov-Perron reducible Markovian master equationhttps://www.zbmath.org/1483.340802022-05-16T20:40:13.078697Z"Szczygielski, Krzysztof"https://www.zbmath.org/authors/?q=ai:szczygielski.krzysztofThe probabilistic point of view on the generalized fractional partial differential equationshttps://www.zbmath.org/1483.350022022-05-16T20:40:13.078697Z"Kolokoltsov, Vassili N."https://www.zbmath.org/authors/?q=ai:kolokoltsov.vassili-nSummary: This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Differential Equations of Caputo and Riemann-Liouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (Monte-Carlo simulation). In many cases these solutions can be compactly presented via the wide class of operator-valued analytic functions of the Mittag-Leffler type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.The Bramson correction for integro-differential Fisher-KPP equationshttps://www.zbmath.org/1483.350282022-05-16T20:40:13.078697Z"Graham, Cole"https://www.zbmath.org/authors/?q=ai:graham.coleSummary: We consider integro-differential Fisher-KPP equations with nonlocal diffusion. For typical equations, we establish the logarithmic Bramson delay for solutions with step-like initial data. That is, these solutions resemble a front at position \(c_\ast t-\frac{3}{2\lambda_\ast} \log t+\mathcal{O}(1)\) for explicit constants \(c_\ast\) and \(\lambda_\ast\). Certain strongly asymmetric diffusions exhibit more exotic behaviour.Asymptotic autonomy of bi-spatial attractors for stochastic retarded Navier-Stokes equationshttps://www.zbmath.org/1483.350442022-05-16T20:40:13.078697Z"Zhang, Qiangheng"https://www.zbmath.org/authors/?q=ai:zhang.qiangheng"Li, Yangrong"https://www.zbmath.org/authors/?q=ai:li.yangrongSummary: We establish semi-convergence of a non-autonomous bi-spatial random attractor towards to an autonomous attractor under the topology of the regular space when time-parameter goes to infinity, where the criteria are given by forward compactness of the attractor in the terminal space as well as forward convergence of the random dynamical system in the initial space. We then apply to both non-autonomous and autonomous stochastic 2D Navier-Stokes equations with general delays (including variable and distribution delays). The forward-pullback asymptotic compactness in the space of continuous Sobolev-valued functions is proved by the method of spectrum decomposition.Chiti-type reverse Hölder inequality and torsional rigidity under integral Ricci curvature conditionhttps://www.zbmath.org/1483.350512022-05-16T20:40:13.078697Z"Chen, Hang"https://www.zbmath.org/authors/?q=ai:chen.hang|chen.hang.1Summary: In this paper, we prove a reverse Hölder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of \textit{N. Gamara} et al. [Open Math. 13, 557--570 (2015; Zbl 06632233)] and \textit{D. Colladay} et al. [J. Geom. Anal. 28, No. 4, 3906--3927 (2018; Zbl 1410.58016)] from the pointwise lower Ricci curvature bound to the integral Ricci curvature condition. We also extend the results from Laplacian to \(p\)-Laplacian.Laplace Dirichlet heat kernels in convex domainshttps://www.zbmath.org/1483.350812022-05-16T20:40:13.078697Z"Serafin, G."https://www.zbmath.org/authors/?q=ai:serafin.grzegorzSummary: We provide general lower and upper bounds for Laplace Dirichlet heat kernel of convex \(\mathcal{C}^{1 , 1}\) domains. The obtained estimates precisely describe the exponential behaviour of the kernels, which has been known only in a few special cases so far. Furthermore, we characterize a class of sets for which the estimates are sharp, i.e. the upper and lower bounds coincide up to a multiplicative constant. In particular, this includes sets of the form \(\{x \in \mathbb{R}^n : x_n > a |( x_1, \ldots, x_{n - 1}) |^p \}\) where \(p \geqslant 2\), \(n \geqslant 2\) and \(a > 0\).A refined Green's function estimate of the time measurable parabolic operators with conic domainshttps://www.zbmath.org/1483.351032022-05-16T20:40:13.078697Z"Kim, Kyeong-Hun"https://www.zbmath.org/authors/?q=ai:kim.kyeonghun"Lee, Kijung"https://www.zbmath.org/authors/?q=ai:lee.kijung"Seo, Jinsol"https://www.zbmath.org/authors/?q=ai:seo.jinsolSummary: We present a refined Green's function estimate of the time measurable parabolic operators on conic domains that involves mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary.Fundamental solution to 1D degenerate diffusion equation with locally bounded coefficientshttps://www.zbmath.org/1483.351242022-05-16T20:40:13.078697Z"Chen, Linan"https://www.zbmath.org/authors/?q=ai:chen.linan"Weih-Wadman, Ian"https://www.zbmath.org/authors/?q=ai:weih-wadman.ianSummary: In this work we study the degenerate diffusion equation \(\partial_t=x^\alpha a(x)\partial_x^2+b(x)\partial_x\) for \((x,t)\in(0,\infty)^2\), equipped with a Cauchy initial data and the Dirichlet boundary condition at 0. We assume that the order of degeneracy at 0 of the diffusion operator is \(\alpha\in(0,2)\), and the coefficient functions (and their derivatives) are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution \(p(x,y,t)\) and prove several properties for \(p(x,y,t)\); by conducting a localization procedure, we obtain an approximation for \(p(x,y,t)\) for \(x,y\) in a neighborhood of 0 and \(t\) sufficiently small, where the error estimates only rely on the local bounds of \(a(x)\) and \(b(x)\) (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of \(\alpha=1\). Our work extends part of the existing results to the cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probabilistic view (e.g., wellposedness of stochastic differential equations).Vlasov-Poisson-Fokker-Planck equation in \(E_{2,1}^s\) spacehttps://www.zbmath.org/1483.352752022-05-16T20:40:13.078697Z"Chen, Jingchun"https://www.zbmath.org/authors/?q=ai:chen.jingchun"He, Cong"https://www.zbmath.org/authors/?q=ai:he.congSummary: In this paper, we are concerned about the local-in-time well-posedness of the Vlasov-Poisson-Fokker-Planck equation in \(E_{2,1}^s (\mathbb{R}^{2n})\) which is a hybrid modulation Lebesgue space and related to the Gevery class only with respect to \(x\) variable. The difficulty lies in the estimates of the electronic term \(\nabla_x \phi \). To handle this, we establish a product formula and \(L^2\)-\(L^q\) estimate.Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamishttps://www.zbmath.org/1483.353062022-05-16T20:40:13.078697Z"Tong, Shanyin"https://www.zbmath.org/authors/?q=ai:tong.shanyin"Vanden-Eijnden, Eric"https://www.zbmath.org/authors/?q=ai:vanden-eijnden.eric"Stadler, Georg"https://www.zbmath.org/authors/?q=ai:stadler.georgSummary: We propose and compare methods for the analysis of extreme events in complex systems governed by PDEs that involve random parameters, in situations where we are interested in quantifying the probability that a scalar function of the system's solution is above a threshold. If the threshold is large, this probability is small and its accurate estimation is challenging. To tackle this difficulty, we blend theoretical results from large deviation theory (LDT) with numerical tools from PDE-constrained optimization. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events, using adjoint methods to compute the gradient of this rate function. The minimizers give information about the mechanism of the extreme events as well as estimates of their probability. We then propose a series of methods to refine these estimates, either via importance sampling or geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided for Gaussian distributions. We give theoretical and numerical arguments showing that the performance of our methods is insensitive to the extremeness of the events we are interested in. We illustrate the application of our approach to quantify the probability of extreme tsunami events on shore. Tsunamis are typically caused by a sudden, unpredictable change of the ocean floor elevation during an earthquake. We model this change as a random process, which takes into account the underlying physics. We use the one-dimensional shallow water equation to model tsunamis numerically. In the context of this example, we present a comparison of our methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore.Approximate solution for a 2-D fractional differential equation with discrete random noisehttps://www.zbmath.org/1483.353312022-05-16T20:40:13.078697Z"Tuan, Nguyen Huy"https://www.zbmath.org/authors/?q=ai:nguyen-huy-tuan."Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-i"Thach, Tran Ngoc"https://www.zbmath.org/authors/?q=ai:thach.tran-ngoc"O'Regan, Donal"https://www.zbmath.org/authors/?q=ai:oregan.donal"Can, Nguyen Huu"https://www.zbmath.org/authors/?q=ai:can.nguyen-huuSummary: We study a boundary value problem for a 2-D fractional differential equation (FDE) with random noise. This problem is not well-posed. Hence, we use truncated regularization method to establish regularized solutions for the such problem. Finally, the convergence rate of this approximate solution and a numerical example are investigated.Solution to a stochastic 3D nonlocal Cahn-Hilliard-Navier-Stokes model with shear dependent viscosity via a splitting-up methodhttps://www.zbmath.org/1483.353482022-05-16T20:40:13.078697Z"Deugoué, G."https://www.zbmath.org/authors/?q=ai:deugoue.gabriel"Moghomye, B. Jidjou"https://www.zbmath.org/authors/?q=ai:moghomye.boris-jidjou|jidjou-moghomye.b"Medjo, T. Tachim"https://www.zbmath.org/authors/?q=ai:tachim-medjo.theodoreSummary: We consider a stochastic version of a nonlinear system, which describes the motion of an incompressible mixture of two immiscible non-Newtonian fluids under the influence of the stochastic external forces. The model consists of the stochastic Navier-Stokes equations with shear dependent viscosity controlled by a power \(p > 2\), coupled with a convective nonlocal Cahn-Hilliard equations. We prove the existence of a weak martingale solution when \(p \ge 11/5\) via a time discretisation based on the splitting-up method.The energy-dissipation principle for stochastic parabolic equationshttps://www.zbmath.org/1483.353492022-05-16T20:40:13.078697Z"Scarpa, Luca"https://www.zbmath.org/authors/?q=ai:scarpa.luca"Stefanelli, Ulisse"https://www.zbmath.org/authors/?q=ai:stefanelli.ulisseSummary: The Energy-Dissipation Principle provides a variational tool for the analysis of parabolic evolution problems: solutions are characterized as so-called null-minimizers of a global functional on entire trajectories. This variational technique allows for applying the general results of the calculus of variations to the underlying differential problem and has been successfully applied in a variety of deterministic cases, ranging from doubly nonlinear flows to curves of maximal slope in metric spaces. The aim of this note is to extend the Energy-Dissipation Principle to stochastic parabolic evolution equations. Applications to stability and optimal control are also presented.Spread out random walks on homogeneous spaceshttps://www.zbmath.org/1483.370072022-05-16T20:40:13.078697Z"Prohaska, Roland"https://www.zbmath.org/authors/?q=ai:prohaska.rolandThe setting considered here is that of a homogeneous space \(X=G/\Gamma\) for a \(\sigma\)-compact locally compact metrizable group \(G\) and a discrete subgroup \(\Gamma<G\), with a Borel probability measure \(\mu\) on \(G\) used to define a random walk on \(X\). That is, each step of the random walk chooses an element \(g\in G\) according to \(\mu\) and then moves \(x\in X\) to \(gx\in X\). Here the Markov chain theory is used to carry out a careful analysis under the assumption that the increment function is spread out. In the lattice (finite volume) case a complete picture of the asymptotics of the \(n\)-step distribution is found, and they are shown to equidistribute to Haar measure. Situations in which this equidistribution is exponentially fast or locally uniform relative to the initial point are studied. In the case of infinite volume the recurrence is shown and it is proved a ratio limit theorem for symmetric spread out random walks on homogeneous spaces under a growth condition.
Reviewer: Thomas B. Ward (Newcastle)Asymptotic decoupling and weak Gibbs measures for finite alphabet shift spaceshttps://www.zbmath.org/1483.370202022-05-16T20:40:13.078697Z"Pfister, C.-E."https://www.zbmath.org/authors/?q=ai:pfister.charles-edouard"Sullivan, W. G."https://www.zbmath.org/authors/?q=ai:sullivan.wayne-gThe paper deals with equilibrium states for continuous functions on a large class of finite-alphabet shift spaces. The authors study the decoupling condition on shift spaces and the space of functions of bounded total oscillations on shift spaces. Their properties and examples are presented. Let \(A\) be a finite set and \(L=\mathbb{Z}^d\). As the main result, the authors prove that if a shift space \(X\subset A^{L}\) satisfies the decoupling condition and \(\phi\) is a function with bounded total oscillations on \(X\), then an equilibrium measure \(\nu\) for \(\phi\) is a weak Gibbs measure for \(\phi-P(\phi)\) where \(P(\phi)\) is the topological pressure of \(\phi\). Then they obtain a full large-deviation principle for the empirical measures on \((X, \nu)\). They prove that if \(X\) is a shift space satisfying the decoupling condition then the ergodic measures on \(X\) are entropy dense. An example of a function of bounded total oscillations not satisfying the Bowen property is provided.
Reviewer: Yuki Yayama (Chiilán)Singularities of invariant densities for random switching between two linear ODEs in 2Dhttps://www.zbmath.org/1483.370622022-05-16T20:40:13.078697Z"Bakhtin, Yuri"https://www.zbmath.org/authors/?q=ai:bakhtin.yuri-yu"Hurth, Tobias"https://www.zbmath.org/authors/?q=ai:hurth.tobias"Lawley, Sean D."https://www.zbmath.org/authors/?q=ai:lawley.sean-d"Mattingly, Jonathan C."https://www.zbmath.org/authors/?q=ai:mattingly.jonathan-cGradient flow of the stochastic relaxation on a generic exponential familyhttps://www.zbmath.org/1483.370632022-05-16T20:40:13.078697Z"Malagò, Luigi"https://www.zbmath.org/authors/?q=ai:malago.luigi"Pistone, Giovanni"https://www.zbmath.org/authors/?q=ai:pistone.giovanniSummary: We study the natural gradient flow of the expected value \(E_p [f]\) of an objective function \(f\) for \(p\) in an exponential family. We parameterize the exponential family with the expectation parameters and we show that the dynamical system associated to the natural gradient flow can be extended outside the marginal polytope.
For the entire collection see [Zbl 1470.00021].A random dynamical systems perspective on isochronicity for stochastic oscillationshttps://www.zbmath.org/1483.370642022-05-16T20:40:13.078697Z"Engel, Maximilian"https://www.zbmath.org/authors/?q=ai:engel.maximilian"Kuehn, Christian"https://www.zbmath.org/authors/?q=ai:kuhn.christianThe authors study the problem of defining isochrons for stochastic oscillations. They propose a new approach for finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. The authors introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows them to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, the relations between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities are discussed.
Reviewer: Chao Wang (Kunming)Optimal control of Clarke subdifferential type fractional differential inclusion with non-instantaneous impulses driven by Poisson jumps and its topological propertieshttps://www.zbmath.org/1483.370942022-05-16T20:40:13.078697Z"Durga, N."https://www.zbmath.org/authors/?q=ai:durga.nagarajan"Muthukumar, P."https://www.zbmath.org/authors/?q=ai:muthukumar.palanisamySummary: This article is devoted to studying the topological structure of a solution set for Clarke subdifferential type fractional non-instantaneous impulsive differential inclusion driven by Poisson jumps. Initially, for proving the solvability result, we use a nonlinear alternative of Leray-Schauder fixed point theorem, Gronwall inequality, stochastic analysis, a measure of noncompactness, and the multivalued analysis. Furthermore, the mild solution set for the proposed problem is demonstrated with nonemptyness, compactness, and, moreover, \(R_\delta\)-set. By employing Balder's theorem, the existence of optimal control is derived. At last, an application is provided to validate the developed theoretical results.Random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations on unbounded domainshttps://www.zbmath.org/1483.370952022-05-16T20:40:13.078697Z"Shu, Ji"https://www.zbmath.org/authors/?q=ai:shu.ji"Zhang, Jian"https://www.zbmath.org/authors/?q=ai:zhang.jianSummary: This paper deals with the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with \(\alpha\in(0,1)\). We prove the existence and uniqueness of tempered pullback random attractors for the equations in \(L^2(\mathbf{R}^3)\). In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in \(L^2(\mathbf{R}^3)\) by the tail-estimates of solutions.Wong-Zakai approximations and attractors for fractional stochastic reaction-diffusion equations on unbounded domainshttps://www.zbmath.org/1483.370962022-05-16T20:40:13.078697Z"Sun, Yaqing"https://www.zbmath.org/authors/?q=ai:sun.yaqing"Gao, Hongjun"https://www.zbmath.org/authors/?q=ai:gao.hongjunSummary: In this paper, we investigate the Wong-Zakai approximations induced by a stationary process and the long term behavior of the fractional stochastic reaction-diffusion equation driven by a white noise. Precisely, one of the main ingredients in this paper is to establish the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of fractional stochastic reaction-diffusion equations. Thereafter the upper semi-continuity of attractors for the Wong-Zakai approximation of the equation as \(\delta\rightarrow0\) is proved.Characterising stochastic fixed points and limit cycles for dynamical systems with additive noisehttps://www.zbmath.org/1483.371012022-05-16T20:40:13.078697Z"Biswas, Saranya"https://www.zbmath.org/authors/?q=ai:biswas.saranya"Rounak, Aasifa"https://www.zbmath.org/authors/?q=ai:rounak.aasifa"Perlikowski, Przemysław"https://www.zbmath.org/authors/?q=ai:perlikowski.przemyslaw"Gupta, Sayan"https://www.zbmath.org/authors/?q=ai:gupta.sayanSummary: This study focuses on characterising numerically the attractor volume in the state space of dynamical systems excited by additive white noise. A definition for stochastic attractors is introduced in terms of probability measure and numerical methodologies are presented to characterise them. The study is limited to investigating the effects of additive noise on fixed point and limit cycle attractors of the corresponding noise free system. The effect of noise intensity on the definition and validity of the proposed methodology is also discussed.Uniqueness of Banach space valued graphonshttps://www.zbmath.org/1483.460102022-05-16T20:40:13.078697Z"Kunszenti-Kovács, Dávid"https://www.zbmath.org/authors/?q=ai:kunszenti-kovacs.davidSummary: A Banach space valued graphon is a function \(W:(\Omega,\mathcal{A}, \pi)^2 \to \mathcal{Z}\) from a probability space to a Banach space with a separable predual, measurable in a suitable sense, and lying in appropriate \(L^p\)-spaces. As such we may consider \(W(x,y)\) as a two-variable random element of the Banach space. A two-dimensional analogue of moments can be defined with the help of graphs and weak-* evaluations, and a natural question that then arises is whether these generalized moments determine the function \(W\) uniquely -- up to measure preserving transformations. The main motivation comes from the theory of multigraph limits, where these graphons arise as the natural limit objects for convergence in a generalized homomorphism sense. Our main result is that this holds true under some Carleman-type condition, but fails in general even with \(\mathcal{Z} = \mathbb{R}\), for reasons related to the classical moment-problem. In particular, limits of multigraph sequences are uniquely determined -- up to measure preserving transformations -- whenever the tails of the edge-distributions stay small enough.RKH spaces of Brownian type defined by Cesàro-Hardy operatorshttps://www.zbmath.org/1483.460232022-05-16T20:40:13.078697Z"Galé, José E."https://www.zbmath.org/authors/?q=ai:gale.jose-e"Miana, Pedro J."https://www.zbmath.org/authors/?q=ai:miana.pedro-j"Sánchez-Lajusticia, Luis"https://www.zbmath.org/authors/?q=ai:sanchez-lajusticia.luisSummary: We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro-Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley-Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.Homomorphisms relative to additive convolutions and max-convolutions: free, Boolean and classical caseshttps://www.zbmath.org/1483.460672022-05-16T20:40:13.078697Z"Hasebe, Takahiro"https://www.zbmath.org/authors/?q=ai:hasebe.takahiro"Ueda, Yuki"https://www.zbmath.org/authors/?q=ai:ueda.yukiSummary: We introduce new homomorphisms relative to additive convolutions and max-convolutions in free, boolean and classical cases. Crucial roles are played by the limit distributions for free multiplicative law of large numbers.Differential equation approximations of stochastic network processes: an operator semigroup approachhttps://www.zbmath.org/1483.471252022-05-16T20:40:13.078697Z"Bátkai, András"https://www.zbmath.org/authors/?q=ai:batkai.andras"Kiss, Istvan Z."https://www.zbmath.org/authors/?q=ai:kiss.istvan-z"Sikolya, Eszter"https://www.zbmath.org/authors/?q=ai:sikolya.eszter"Simon, Péter L."https://www.zbmath.org/authors/?q=ai:simon.peter-lSummary: The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view, the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size (\(N\)). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as \(N\) tends to infinity. Using only elementary semigroup theory, we can prove the order \(\mathcal{O}(1/N)\) convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a~new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.Multigrid preconditioners for optimal control problems with stochastic elliptic PDE constraintshttps://www.zbmath.org/1483.490282022-05-16T20:40:13.078697Z"Soane, Ana Maria"https://www.zbmath.org/authors/?q=ai:soane.ana-mariaSummary: In this work, we construct multigrid preconditioners to be used in the solution process of pathwise optimal control problems constrained by elliptic partial differential equations with random coefficients. We use a sparse-grid collocation method to discretize in the stochastic space and multigrid techniques in the physical space. Numerical results show that the proposed preconditioners lead to significant computational savings, with the number of preconditioned conjugate gradient iterations decreasing as the resolution in the physical space increases.Mean-field limit for a class of stochastic ergodic control problemshttps://www.zbmath.org/1483.490522022-05-16T20:40:13.078697Z"Albeverio, Sergio"https://www.zbmath.org/authors/?q=ai:albeverio.sergio-a"De Vecchi, Francesco C."https://www.zbmath.org/authors/?q=ai:de-vecchi.francesco-carlo"Romano, Andrea"https://www.zbmath.org/authors/?q=ai:romano.andrea"Ugolini, Stefania"https://www.zbmath.org/authors/?q=ai:ugolini.stefaniaLarge planar Poisson-Voronoi cells containing a given convex bodyhttps://www.zbmath.org/1483.520022022-05-16T20:40:13.078697Z"Calka, Pierre"https://www.zbmath.org/authors/?q=ai:calka.pierre"Demichel, Yann"https://www.zbmath.org/authors/?q=ai:demichel.yann"Enriquez, Nathanaël"https://www.zbmath.org/authors/?q=ai:enriquez.nathanaelSummary: Let \(K\) be a convex body in \(\mathbb{R}^2\). We consider the Voronoi tessellation generated by a homogeneous Poisson point process of intensity \(\lambda\) conditional on the existence of a cell \(K_\lambda\) which contains \(K\). When \(\lambda\rightarrow\infty\), this cell \(K_\lambda\) converges \textit{from above} to \(K\) and we provide the precise asymptotics of the expectation of its defect area, defect perimeter and number of vertices. As in Rényi and Sulanke's seminal papers on random convex hulls, the regularity of \(K\) has crucial importance and we deal with both the smooth and polygonal cases. Techniques are based on accurate estimates of the area of the Voronoi flower and of the support function of \(K_\lambda\) as well as on an Efron-type relation. Finally, we show the existence of limiting variances in the smooth case for the defect area and the number of vertices as well as analogous expectation asymptotics for the so-called Crofton cell.The \(\beta\)-Delaunay tessellation. III: Kendall's problem and limit theorems in high dimensionshttps://www.zbmath.org/1483.520032022-05-16T20:40:13.078697Z"Gusakova, Anna"https://www.zbmath.org/authors/?q=ai:gusakova.anna"Kabluchko, Zakhar"https://www.zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Thäle, Christoph"https://www.zbmath.org/authors/?q=ai:thale.christophThis paper is the third one in a series about the \(\beta\)-Delaunay tessellation (numbers 1,2,4 are available at \url{arXiv:2005.13875v2}, \url{arXiv:2101.11316}, \url{arXiv.2108.09472}). The construction of such a tessellation is based on a Poisson point process in the space \({\mathbb R}^{d-1}\times [0,\infty)\). Its intensity measure \(\mu_\beta\) has, with respect to Lebesgue measure, a density of the form \((v,h)\mapsto \gamma c_{d,\beta}\cdot h^\beta\), where \(\beta>-1\), \(\gamma>0\), and \(c_{d,\beta}\) is a normalizing constant. The `paraboloid hull process' is then used to derive a stationary random tessellation \({\mathcal D}_\beta\) of \({\mathbb R}^{d-1}\) into simplices (alternatively, Laguerre tessellations can be employed to construct \({\mathcal D}_\beta\)). The classical Poisson--Delaunay tessellation can be considered as the limit case when \(\beta\to -1\). The \(\nu\)-weighted typical cell \(Z_{\beta,\nu}\) of \({\mathcal D}_\beta\), weighted by the \(\nu\)th power of the volume (for some \(\nu\ge -1\)), is introduced by using Palm calculus for marked point processes.
The first part of this paper treats Kendall's problem for the \(\nu\)-weighted typical cell: under the condition that \(Z_{\beta,\nu}\) has large volume, an estimate is given showing that the probability that \(Z_{\beta,\nu}\) has large deviation from a regular simplex must be small. Also in this part, it is shown that \({\mathbb P}(\mathrm{Vol}(Z_{\beta,\nu})\le a)\) behaves like a power of \(a\) as \(a\to 0\), and that it decays exponentially as \(a\to\infty\).
The last part of the paper deals with the asymptotic behavior of \(\log\mathrm{Vol}(Z_{\beta,\nu})\) as \(d+2\beta+\nu\to\infty\) (with a particular view to case where \(\beta,\nu\) are constant and the dimension tends to infinity). The cumulant method and mod-\(\phi\) convergence are used. Obtained are a central limit theorem with Berry-Esseen bound, a moderate deviation principle, concentration inequalities, and large deviation behavior.
Reviewer: Rolf Schneider (Freiburg im Breisgau)The volume of simplices in high-dimensional Poisson-Delaunay tessellationshttps://www.zbmath.org/1483.520042022-05-16T20:40:13.078697Z"Gusakova, Anna"https://www.zbmath.org/authors/?q=ai:gusakova.anna"Thäle, Christoph"https://www.zbmath.org/authors/?q=ai:thale.christophSummary: Typical weighted random simplices \(Z_\mu\), \(\mu \in (-2, \infty)\), in a Poisson-Delaunay tessellation in \(\mathbb{R}^n\) are considered, where the weight is given by the \((\mu +1)\)st power of the volume. As special cases this includes the typical (\(\mu =-1\)) and the usual volume-weighted (\(\mu =0\)) Poisson-Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of \(Z_\mu\) satisfies a central limit theorem in high dimensions, that is, as \(n\rightarrow \infty\). In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight \(\mu =\mu(n)\) to depend on the dimension \(n\) as well. A number of special cases are discussed separately. For fixed \(\mu\) also mod-\(\phi\) convergence and the large deviations behaviour of the logarithmic volume of \(Z_\mu\) are investigated.Monochromatic random waves for general Riemannian manifoldshttps://www.zbmath.org/1483.530032022-05-16T20:40:13.078697Z"Canzani, Yaiza"https://www.zbmath.org/authors/?q=ai:canzani.yaizaSummary: This is a survey article on some of the recent developments on monochromatic random waves defined for general Riemannian manifolds. We discuss the conditions needed for the waves to have a universal scaling limit, we review statistics for the size of their zero set and the number of their critical points, and we discuss the structure of their zero set as described by the diffeomorphism types and the nesting configurations of its components.
For the entire collection see [Zbl 1473.53004].Stochastic test of a minimal surfacehttps://www.zbmath.org/1483.530062022-05-16T20:40:13.078697Z"Klimentov, D. S."https://www.zbmath.org/authors/?q=ai:klimentov.dmitrii-sergeevichSummary: The paper aims to obtain a stochastic test for minimal surfaces. Such a test is formulated in terms of transition densities of stochastic processes. Two fundamental forms of the surface generate these processes. This work exhausts the problem of stochastic test for regular minimal surfaces.
For the entire collection see [Zbl 1470.47003].Statistics on Lie groups: a need to go beyond the pseudo-Riemannian frameworkhttps://www.zbmath.org/1483.530262022-05-16T20:40:13.078697Z"Miolane, Nina"https://www.zbmath.org/authors/?q=ai:miolane.nina"Pennec, Xavier"https://www.zbmath.org/authors/?q=ai:pennec.xavierSummary: Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group \(G\) is a manifold that carries an additional group structure. Statistics on \textit{Riemannian} manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall by others. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is \textit{compatible with the group structure}, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group \(G\). The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.
For the entire collection see [Zbl 1470.00021].Poisson-Voronoi tessellation on a Riemannian manifoldhttps://www.zbmath.org/1483.530332022-05-16T20:40:13.078697Z"Calka, Pierre"https://www.zbmath.org/authors/?q=ai:calka.pierre"Chapron, Aurélie"https://www.zbmath.org/authors/?q=ai:chapron.aurelie"Enriquez, Nathanaël"https://www.zbmath.org/authors/?q=ai:enriquez.nathanaelThis paper pioneers the study of the Poisson-Voronoi tessellation on Riemannian manifolds from the perspective of stochastic geometry.
Classically, the Poisson-Voronoi tessellation is defined on Euclidean spaces as follows: first, start with a Poisson point process \(\mathcal{P}_\lambda\) with intensity \(\lambda\). Then, associate each point \(x \in \mathbb{R}^d\) to its nearest neighbor in \(\mathcal{P}\). Thus, each point \(p \in \mathcal{P}\) is associated with a cell consisting of all points closest to it
\[ C(p,\mathcal{P}_\lambda) = \{x \in \mathbb{R}^d: d(x,p) \leq d(x,p')\, \forall p' \in \mathcal{P}\}. \]
The collection of all such cell forms a tessellation of \(\mathbb{R}^d\), called the Poisson-Voronoi tessellation. For Euclidean spaces, \(d\) is the Euclidean distance. In this paper, \(\mathbb{R}^d\) is replaced by a Riemannian manifold \(M\), and \(d\) is the Riemannian metric of \(M\).
Typical results in stochastic geometry concern statistics of the tessellation. This paper focuses on statistics of the typical cell. The typical cell \(\mathcal{C}^{M}_{x_0,\lambda}\) is defined as the cell that contains \(x_0\) when the \(x_0\) is added to \(\mathcal{P}_\lambda\). Intuitively, it captures the local geometry of the tessellation around a point \(x_0 \in M\). The main results of this paper include high-intensity asymptotics for the mean number of vertices and density of vertices of the typical cell \(\mathcal{C}^{M}_{x_0,\lambda}\), under some curvature assumptions on \(M\). Results on these quantities have previously been obtained only for Euclidean spaces, and for two non-Euclidean manifolds only, namely the sphere and the hyperbolic space.
Reviewer: Ngoc Mai Tran (Bonn)A note on the Gannon-Lee theoremhttps://www.zbmath.org/1483.530392022-05-16T20:40:13.078697Z"Schinnerl, Benedict"https://www.zbmath.org/authors/?q=ai:schinnerl.benedict"Steinbauer, Roland"https://www.zbmath.org/authors/?q=ai:steinbauer.rolandSummary: We prove a Gannon-Lee theorem for non-globally hyperbolic Lorentzian metrics of regularity \(C^1\), the most general regularity class currently available in the context of the classical singularity theorems. Along the way, we also prove that any maximizing causal curve in a \(C^1\)-spacetime is a geodesic and hence of \(C^2\)-regularity.Iterated Brownian motion ad libitum is not the pseudo-archttps://www.zbmath.org/1483.540182022-05-16T20:40:13.078697Z"Casse, Jérôme"https://www.zbmath.org/authors/?q=ai:casse.jerome"Curien, Nicolas"https://www.zbmath.org/authors/?q=ai:curien.nicolasSummary: The construction of a random continuum \(\mathcal{C}\) from independent two-sided Brownian motions as considered in [\textit{V. Kiss} and \textit{S. Solecki}, Bull. Lond. Math. Soc. 53, No. 5, 1376--1389 (2021; Zbl 07456946)] almost surely yields a non-degenerate indecomposable continuum. We show that \(\mathcal{C}\) is not-hereditarily indecomposable and, in particular, it is (unfortunately) not the pseudo-arc.Renormalization of stochastic continuity equations on Riemannian manifoldshttps://www.zbmath.org/1483.580092022-05-16T20:40:13.078697Z"Galimberti, Luca"https://www.zbmath.org/authors/?q=ai:galimberti.luca"Karlsen, Kenneth H."https://www.zbmath.org/authors/?q=ai:karlsen.kenneth-hvistendahlThe authors study initial value problems for stochastic continuity equations on smooth closed Riemannian manifolds \(M\) with metric \(h\), of the form \[ \partial_{t}\rho + \operatorname{div}_{h}\left[ \rho \left( u(t,x) + \sum_{i=1}^{N} a_{i}(x) \circ \frac{d W^{i}}{dt} \right) \right]=0, \tag{1} \] for Sobolev velocity fields \(u\), perturbed by Gaussian noise terms driven by that independent Wiener processes \(W^{i}\), where \(a_{i}\) are smooth spatially dependent vector fields on M (with the stochastic integrals interpreted in the Stratonovich sense), supplemented with initial data \(\rho(0)=\rho_{0} \in L^2(M)\).
This type of equation is very interesting both from the mathematical point of view as well as from the point of view of applications (e.g. in fluid mechanics). The deterministic case (\(a_{i}=0\)) has been studied and existence of weak solution was shown using the DiPerna-Lions theory of renormalized solutions [\textit{R. J. DiPerna} and \textit{P. L. Lions}, Invent. Math. 98, No. 3, 511--547 (1989; Zbl 0696.34049)], both in the Euclidean and the smooth closed manifold case and there are important extensions by \textit{L. Ambrosio} [ibid. 158, No. 2, 227--260 (2004; Zbl 1075.35087)] in the case of BV velocity fields). We recall that a renormalized solution \(\rho\) is a weak solution such that \(S(\rho)\) is also a weak solution for any ``reasonable'' \(S : {\mathbb R} \to {\mathbb R}\). The stochastic case for Lipschitz type coefficients has been studied by \textit{H. Kunita} [Stochastic flows and stochastic differential equations. Cambridge etc.: Cambridge University Press (1990; Zbl 0743.60052)] in the Euclidean case, whereas results by \textit{S. Attanasio} and \textit{F. Flandoli} [Commun. Partial Differ. Equations 36, No. 7--9, 1455--1474 (2011; Zbl 1237.60048)] establish the renormalization property for BV velocity field and constant \(a_{i}\), revealing an interesting regularization by noise property of the equation (in the sense that the renormalization property implies uniqueness without the usual \(L^{\infty}\) assumption on the divergence of \(u\)) which has become a recurrent theme in the study of stochastic transport or continuity equations. Extensions in the case of stochastic continuity equations in Itō form in the Euclidean domain with spatially dependent noise coefficients were obtained in [\textit{S. Punshon-Smith} and \textit{S. Smith}, Arch. Ration. Mech. Anal. 229, No. 2, 627--708 (2018; Zbl 1394.35313)] and [\textit{J. A. Rossmanith} et al., J. Comput. Phys. 199, No. 2, 631--662 (2004; Zbl 1126.76350)].
In this paper the authors study problem (1) in the generalized setting already mentioned above, and their main result is the renormalization property for weak \(L^2\) solutions of (1). As a corollary they deduce the uniqueness of weak solutions and an a priori estimate under the usual condition that \(\operatorname{div}_{h} u \in L_{t}^{1}L^{\infty}\). The proof consists of a systematic procedure for regularizing tensor fields on a manifold, a convenient choice of atlas to simplify technical computations linked to the Christoffel symbols, and several DiPerna-Lions type commutators between (first/second order) geometric differential operators and the regularization device.
Reviewer: Athanasios Yannacopoulos (Athína)Markov renewal and piecewise deterministic processeshttps://www.zbmath.org/1483.600012022-05-16T20:40:13.078697Z"Cocozza-Thivent, Christiane"https://www.zbmath.org/authors/?q=ai:cocozza-thivent.christianeThe book offers a modern and comprehensive presentation of the \textit{piecewise deterministic Markov processes} (PDMPs) with general interarrival distributions. The interest for the study of this topic is very well motivated by a large number of applications in various fields: systems reliability, economics and finance, engineering, physics, geology and others. According to the author, the conception of this work is based on the critical thinking of concrete models. Thus, the book complements the monograph [\textit{M. H. A. Davis}, Markov models and optimization. London: Chapman \& Hall (1993; Zbl 0780.60002)]. \par The introduction in the topic is made by the description of the Markov renewal processes and of some associated processes (Chapter 2). A general PDMP is defined in Chapter 3 as a function of a \textit{completed semi-Markov process} (CSMP). The corresponding parameterized processes are discussed and customized. Chapter 4 develops two methods for estimating the probability distribution of hitting times. The intensities of marked point processes are studied and exemplified in Chapter 5. The Kolmogorov equations and the martingales related to CSMPs and PDMPs are highlighted in Chapters 6 and 7. A wealth of information about PDMP is provided by the study of the following topics: stability (Chapter 8), numerical methods (Chapter 9), and switching processes (Chapter 10). \par Reading the book is instructive and enjoyable. The author, with a rich experience in the field, proposes us new perspectives on the Markov processes and their applications.
Reviewer: Eugen Paltanea (Braşov)Markovian arrival processes in multi-dimensionshttps://www.zbmath.org/1483.600022022-05-16T20:40:13.078697Z"Blume, Andreas"https://www.zbmath.org/authors/?q=ai:blume.andreas"Buchholz, Peter"https://www.zbmath.org/authors/?q=ai:buchholz.peter"Scherbaum, Clara"https://www.zbmath.org/authors/?q=ai:scherbaum.claraSummary: Phase Type Distributions (PHDs) and Markovian Arrival Processes (MAPs) are established models in computational probability to describe random processes in stochastic models. In this paper we extend MAPs to Multi-Dimensional MAPs (MDMAPs) which are a model for random vectors that may be correlated in different dimensions. The computation of different quantities like joint moments or conditional densities is introduced and a first approach to compute parameters with respect to measured data is presented.
For the entire collection see [Zbl 1475.68022].Convex algebras of probability distributions induced by finite associative ringshttps://www.zbmath.org/1483.600032022-05-16T20:40:13.078697Z"Yashunsky, Alexey D."https://www.zbmath.org/authors/?q=ai:yashunsky.aleksey-dSummary: We consider the transformations of random variables over a finite associative ring by the addition and multiplication operations. For arbitrary finite rings, we construct families of distribution algebras, which are sets of distributions closed over sums and products of independent random variables.Almost all non-archimedean Kakeya sets have measure zerohttps://www.zbmath.org/1483.600042022-05-16T20:40:13.078697Z"Caruso, Xavier"https://www.zbmath.org/authors/?q=ai:caruso.xavierSummary: We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets are neglectable according to the Haar measure. We also discuss possible relations with the non-archimedean Kakeya conjecture.Information measures in records and their concomitants arising from Sarmanov family of bivariate distributionshttps://www.zbmath.org/1483.600052022-05-16T20:40:13.078697Z"Husseiny, I. A."https://www.zbmath.org/authors/?q=ai:husseiny.islam-a"Barakat, H. M."https://www.zbmath.org/authors/?q=ai:barakat.haroon-mohamed"Mansour, G. M."https://www.zbmath.org/authors/?q=ai:mansour.g-m"Alawady, M. A."https://www.zbmath.org/authors/?q=ai:alawady.metwally-alsayedSummary: One of the most pliable and robust extensions of the classical FGM family of bivariate distributions is the Sarmanov family, which was proposed and used by Sarmanov (1974) as a new model of hydrological processes, inter alia. Despite the salient and almost unique features of this family, it is never used in the literature. The distribution theory of concomitants of record values from this family is investigated. Furthermore, the joint distribution of concomitants of record values for this family is studied. Besides, some aspects of information measures, namely, the Shannon entropy, inaccuracy measure, extropy, cumulative entropy, and Fisher information number are studied. Illustrative examples are provided, where numerical studies lend further support to the theoretical results.On a distinguished family of random variables and Painlevé equationshttps://www.zbmath.org/1483.600062022-05-16T20:40:13.078697Z"Assiotis, Theodoros"https://www.zbmath.org/authors/?q=ai:assiotis.theodoros"Bedert, Benjamin"https://www.zbmath.org/authors/?q=ai:bedert.benjamin"Gunes, Mustafa Alper"https://www.zbmath.org/authors/?q=ai:gunes.mustafa-alper"Soor, Arun"https://www.zbmath.org/authors/?q=ai:soor.arunSummary: A family of random variables \(\mathbf{X}(s)\), depending on a real parameter \(s>-\frac{1}{2} \), appears in the asymptotics of the joint moments of characteristic polynomials of random unitary matrices and their derivatives [\textit{T. Assiotis} et al., ``On the joint moments of the characteristic polynomials of random unitary matrices'', Preprint, \url{arXiv:2005.13961}], in the ergodic decomposition of the Hua-Pickrell measures [\textit{A. Borodin} and \textit{G. Olshanski}, Commun. Math. Phys. 223, No. 1, 87--123 (2001; Zbl 0987.60020); \textit{Y. Qiu}, Adv. Math. 308, 1209--1268 (2017; Zbl 1407.60011], and conjecturally in the asymptotics of the joint moments of Hardy's function and its derivative ([\textit{C. Hughes}, On the characteristic polynomial of a random unitary matrix and the Riemann zeta function. Heslington: University of York (PhD Thesis) (2001)] and [Assiotis et al., loc. cit.]). Our first main result establishes a connection between the characteristic function of \(\mathbf{X}(s)\) and the \(\sigma\)-Painlevé \(\text{III}^\prime\) equation in the full range of parameter values \(s>-\frac{1}{2} \). Our second main result gives the first explicit expression for the density and all the complex moments of the absolute value of \(\mathbf{X}(s)\) for integer values of \(s\). Finally, we establish an analogous connection to another special case of the \(\sigma \)-Painlevé \(\text{III}^\prime\) equation for the Laplace transform of the sum of the inverse points of the Bessel point process.Universality of approximate message passing algorithmshttps://www.zbmath.org/1483.600072022-05-16T20:40:13.078697Z"Chen, Wei-Kuo"https://www.zbmath.org/authors/?q=ai:chen.wei-kuo"Lam, Wai-Kit"https://www.zbmath.org/authors/?q=ai:lam.wai-kitSummary: We consider a broad class of Approximate Message Passing (AMP) algorithms defined as a Lipschitzian functional iteration in terms of an \(n\times n\) random symmetric matrix \(A\). We establish universality in noise for this AMP in the \(n\)-limit and validate this behavior in a number of AMPs popularly adapted in compressed sensing, statistical inferences, and optimizations in spin glasses.Tail bounds for gaps between eigenvalues of sparse random matriceshttps://www.zbmath.org/1483.600082022-05-16T20:40:13.078697Z"Lopatto, Patrick"https://www.zbmath.org/authors/?q=ai:lopatto.patrick"Luh, Kyle"https://www.zbmath.org/authors/?q=ai:luh.kyleThe authors study the eigenvalue gaps of sparse random matrices. The theory of sparse random matrices is of interest in its own right, but it also has innumerable applications in computer science and statistics. In contexts where sparse random matrices have similar spectral guarantees as their dense counterparts, they offer significant advantages as they require less space to store, allow quicker multiplication, and fewer random bits is necessary to generate them. The main contribution of the paper is to go beyond verifying the fact that such matrices have simple spectrum, and prove a tail bound for the minimal eigenvalue gap of these sparse random matrices. So, the first eigenvalue repulsion bound for sparse random matrices is established. As a consequence, it is proved that these matrices have simple spectrum, improving the range of sparsity and error probability from the previous works. It is also proved that for sparse Erdős-Rényi graphs, weak and strong nodal domains are the same.
Reviewer: Yuliya S. Mishura (Kyïv)On the largest common subtree of random leaf-labeled binary treeshttps://www.zbmath.org/1483.600092022-05-16T20:40:13.078697Z"Aldous, David J."https://www.zbmath.org/authors/?q=ai:aldous.david-jPeriodic Pólya urns and an application to Young tableauxhttps://www.zbmath.org/1483.600102022-05-16T20:40:13.078697Z"Banderier, Cyril"https://www.zbmath.org/authors/?q=ai:banderier.cyril"Marchal, Philippe"https://www.zbmath.org/authors/?q=ai:marchal.philippe"Wallner, Michael"https://www.zbmath.org/authors/?q=ai:wallner.michaelSummary: Pólya urns are urns where at each unit of time a ball is drawn and is replaced with some other balls according to its colour. We introduce a more general model: The replacement rule depends on the colour of the drawn ball and the value of the time \(\pmod p\). We discuss some intriguing properties of the differential operators associated to the generating functions encoding the evolution of these urns. The initial nonlinear partial differential equation indeed leads to linear differential equations and we prove that the moment generating functions are D-finite. For a subclass, we exhibit a closed form for the corresponding generating functions (giving the exact state of the urns at time \(n\)). When the time goes to infinity, we show that these periodic Pólya urns follow a rich variety of behaviours: their asymptotic fluctuations are described by a family of distributions, the generalized Gamma distributions, which can also be seen as powers of Gamma distributions. En passant, we establish some enumerative links with other combinatorial objects, and we give an application for a new result on the asymptotics of Young tableaux: This approach allows us to prove that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions.
For the entire collection see [Zbl 1390.68020].A time-dependent Pólya urn with multiple drawingshttps://www.zbmath.org/1483.600112022-05-16T20:40:13.078697Z"Chen, May-Ru"https://www.zbmath.org/authors/?q=ai:chen.mayruSummary: In this paper, we consider a generalized Pólya urn model with multiple drawings and time-dependent reinforcements. Suppose an urn initially contains \(w\) white and \(r\) red balls. At the \(n\)th action, \(m\) balls are drawn at random from the urn, say \(k\) white and \(m-k\) red balls, and then replaced in the urn along with \(c_nk\) white and \(c_n(m-k)\) red balls, where \(\{c_n\}\) is a given sequence of positive integers. Repeat the above procedure \textit{ad infinitum}. Let \(X_n\) be the proportion of the white balls in the urn after the \(n\)th action. We first show that \(X_n\) converges almost surely to a random variable \(X\). Next, we give a necessary and sufficient condition for \(X\) to have a Bernoulli distribution with parameter \(w/(w+r)\). Finally, we prove that \(X\) is absolutely continuous if \(\{c_n\}\) is bounded.Eigenvalues of the non-backtracking operator detached from the bulkhttps://www.zbmath.org/1483.600122022-05-16T20:40:13.078697Z"Coste, Simon"https://www.zbmath.org/authors/?q=ai:coste.simon"Zhu, Yizhe"https://www.zbmath.org/authors/?q=ai:zhu.yizheDistribution of the number of corners in tree-like and permutation tableauxhttps://www.zbmath.org/1483.600132022-05-16T20:40:13.078697Z"Hitczenko, Pawel"https://www.zbmath.org/authors/?q=ai:hitczenko.pawel"Yaroslavskiy, Aleksandr"https://www.zbmath.org/authors/?q=ai:yaroslavskiy.aleksandrSummary: In this abstract, we study tree-like tableaux and some of their probabilistic properties. Tree-like tableaux are in bijection with other combinatorial structures, including permutation tableaux, and have a connection to the partially asymmetric simple exclusion process (PASEP), an important model of interacting particles system. In particular, in the context of tree-like tableaux, a corner corresponds to a node occupied by a particle that could jump to the right while inner corners indicate a particle with an empty node to its left. Thus, the total number of corners represents the number of nodes at which PASEP can move, i.e., the total current activity of the system. As the number of inner corners and regular corners is connected, we limit our discussion to just regular corners and show that, asymptotically, the number of corners in a tableaux of length \(n\) is normally distributed. Furthermore, since the number of corners in tree-like tableaux are closely related to the number of corners in permutation tableaux, we will discuss the corners in the context of the latter tableaux.
For the entire collection see [Zbl 1390.68020].Patterns in random permutations avoiding some other patternshttps://www.zbmath.org/1483.600142022-05-16T20:40:13.078697Z"Janson, Svante"https://www.zbmath.org/authors/?q=ai:janson.svanteSummary: Consider a random permutation drawn from the set of permutations of length \(n\) that avoid a given set of one or several patterns of length 3. We show that the number of occurrences of another pattern has a limit distribution, after suitable scaling. In several cases, the limit is normal, as it is in the case of unrestricted random permutations; in other cases the limit is a non-normal distribution, depending on the studied pattern. In the case when a single pattern of length 3 is forbidden, the limit distributions can be expressed in terms of a Brownian excursion.\par The analysis is made case by case; unfortunately, no general method is known, and no general pattern emerges from the results.
For the entire collection see [Zbl 1390.68020].Large deviations of convex hulls of planar random walks and Brownian motionshttps://www.zbmath.org/1483.600152022-05-16T20:40:13.078697Z"Akopyan, Arseniy"https://www.zbmath.org/authors/?q=ai:akopyan.arseniy-v"Vysotsky, Vladislav"https://www.zbmath.org/authors/?q=ai:vysotsky.vladislav-vSummary: We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments.
We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments. However, line segments may not be optimal in general.
Furthermore, we find explicitly the rate function of the area of the convex hull for random walks with rotationally invariant distribution of increments. For such walks, which necessarily have zero mean, large deviations of the area are attained by the trajectories that asymptotically align into half-circles. For random walks with non-zero mean increments, we find the rate function of the area for Gaussian walks with drift. Here the optimal limit shapes are elliptic arcs if the covariance matrix of increments is non-degenerate and parabolic arcs if otherwise.
The above results on convex hulls of Gaussian random walks remain valid for convex hulls of planar Brownian motions of all possible parameters. Moreover, we extend the LDPs for the perimeter and the area of convex hulls to general Lévy processes with finite Laplace transform.Unimodular Hausdorff and Minkowski dimensionshttps://www.zbmath.org/1483.600162022-05-16T20:40:13.078697Z"Baccelli, François"https://www.zbmath.org/authors/?q=ai:baccelli.francois-louis"Haji-Mirsadeghi, Mir-Omid"https://www.zbmath.org/authors/?q=ai:haji-mirsadeghi.mir-omid"Khezeli, Ali"https://www.zbmath.org/authors/?q=ai:khezeli.aliSummary: This work introduces two new notions of dimension, namely the \textit{unimodular Minkowski and Hausdorff dimensions}, which are inspired from the classical analogous notions. These dimensions are defined for \textit{unimodular discrete spaces}, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley's lemma and Frostman's lemma are presented. These last lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff measure, and unimodular dimension function are also introduced. This toolbox allows one to connect the unimodular dimensions to other notions such as volume growth rate, discrete dimension and scaling limits. It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces.A nonamenable ``factor'' of a Euclidean spacehttps://www.zbmath.org/1483.600172022-05-16T20:40:13.078697Z"Timár, Ádám"https://www.zbmath.org/authors/?q=ai:timar.adamSummary: Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space \({\mathbb{R}^d}\), \(d\ge 3\), into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in \({\mathbb{R}^d}\) as an isometry-invariant random partition of \({\mathbb{R}^d}\) to bounded polyhedra, and also as an isometry-invariant random partition of \({\mathbb{R}^d}\) to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of IID's.The zero-truncated Poisson-weighted exponential distribution with applicationshttps://www.zbmath.org/1483.600182022-05-16T20:40:13.078697Z"Atikankul, Yupapin"https://www.zbmath.org/authors/?q=ai:atikankul.yupapin"Thongteeraparp, Ampai"https://www.zbmath.org/authors/?q=ai:thongteeraparp.ampai"Bodhisuwan, Winai"https://www.zbmath.org/authors/?q=ai:bodhisuwan.winai"Qin, Jin"https://www.zbmath.org/authors/?q=ai:qin.jin"Boonto, Sudchai"https://www.zbmath.org/authors/?q=ai:boonto.sudchaiSummary: In this article, a new distribution family for non-zero count data is proposed. It is developed from the Poisson-weighted exponential distribution. Various theoretical properties of the proposed distribution, such as the probability generating function, moment generating function, characteristic function, and moments are discussed. The method of maximum likelihood is used to estimate the parameters. Finally, some real data sets are applied to show the performance of the proposed distribution.Zero-truncated negative binomial weighted-Lindley distribution and its applicationhttps://www.zbmath.org/1483.600192022-05-16T20:40:13.078697Z"Bodhisuwan, Rujira"https://www.zbmath.org/authors/?q=ai:bodhisuwan.rujira"Denthet, Sunthree"https://www.zbmath.org/authors/?q=ai:denthet.sunthree"Acoose, Tannen"https://www.zbmath.org/authors/?q=ai:acoose.tannenSummary: In this article, a zero-truncated version of the negative binomial-weighted Lindley distribution is developed, it can be used when the response variable is a count and cannot be zero, it is so-called zero truncated distribution. Some probability characteristics of the proposed distribution are discussed. Random variate generate of the zero-truncated negative binomial-weighted Lindley distribution is presented. The model parameters are estimated by using the maximum likelihood estimation. Moreover, the application study is illustrated by using real data. Efficiency of data fitting among the proposed distribution and the existing distribution are illustrated.A new method for generalizing Burr and related distributionshttps://www.zbmath.org/1483.600202022-05-16T20:40:13.078697Z"Chakraborty, Tanujit"https://www.zbmath.org/authors/?q=ai:chakraborty.tanujit"Das, Suchismita"https://www.zbmath.org/authors/?q=ai:das.suchismita"Chattopadhyay, Swarup"https://www.zbmath.org/authors/?q=ai:chattopadhyay.swarupSummary: A new method has been proposed to generalize Burr-XII distribution, also called Burr distribution, by adding an extra parameter to an existing Burr distribution for more flexibility. In this method, the exponent of the Burr distribution is modeled using a nonlinear function of the data and one additional parameter. The models of this newly introduced generalized Burr family can significantly increase the flexibility of the former Burr distribution with respect to the density and hazard rate shapes. Families expanded using the method proposed here is heavy-tailed and belongs to the maximum domain of attractions of the Frechet distribution. The method is further applied to yield three-parameter classical Pareto and generalized exponentiated distributions which shows the broader application of the proposed idea of generalization. A relevant model of the new generalized Burr family has been considered in detail, with particular emphasis on the hazard functions, stochastic orders, estimation procedures, and testing methods are derived. Finally, as empirical evidence, the new distribution is applied to the analysis of large-scale heavy-tailed network data and compared with other commonly used distributions available for fitting degree distributions of networks. Experimental results suggest that the proposed Burr distribution with nonlinear exponent better fits the large-scale heavy-tailed networks better than the popularly used Marhsall-Olkin generalization of Burr and exponentiated Burr distributions.A new mixed negative binomial Kumaraswamy-Lindley distribution and its applicationhttps://www.zbmath.org/1483.600212022-05-16T20:40:13.078697Z"Denthet, Sunthree"https://www.zbmath.org/authors/?q=ai:denthet.sunthree"Ngamkham, Thuntida"https://www.zbmath.org/authors/?q=ai:ngamkham.thuntidaSummary: In this paper, a new discrete distribution have been proposed, which is the mixed distribution between the negative binomial (NB) distribution and Kumaraswamy-Lindley (KL) distribution called the negative binomial-Kumaraswamy Lindley (NB-KL) distribution. The NB-KL distribution has a special case, that is, a negative binomial weighted Lindley distribution. Some properties of the NB-KL distribution are discussed, e.g., mean, variance, factorial moment. Also, the graphical representations related to the probability mass function of the as NB-KL distribution is provided. We conduct a simulation study and the criterion in parameter estimator comparison is the root mean square error. In application study, the Komogorov-Smirnov goodness of fit test based on the Poisson and negative binomial distributions are discussed. The result emphasize that the NB-KL distribution can be considered as a competitive distribution for count data analysis.The U family of distributions: properties and applicationshttps://www.zbmath.org/1483.600222022-05-16T20:40:13.078697Z"Jamal, Farrukh"https://www.zbmath.org/authors/?q=ai:jamal.farrukh"Chesneau, Christophe"https://www.zbmath.org/authors/?q=ai:chesneau.christophe"Saboor, Abdus"https://www.zbmath.org/authors/?q=ai:saboor.abdus"Aslam, Muhammad"https://www.zbmath.org/authors/?q=ai:aslam.muhammad-saeed|aslam.muhammad-zubair|aslam.muhammad-shamrooz|aslam.muhammad-jamil|aslam.muhammad-kamran|aslam.muhammad-nauman"Tahir, Muhammad H."https://www.zbmath.org/authors/?q=ai:tahir.muhammad-hussain"Mashwani, Wali Khan"https://www.zbmath.org/authors/?q=ai:mashwani.wali-khanSummary: In this article, we develop a new general family of distributions aimed at unifying some well-established lifetime distributions and offering new work perspectives. A special family member based on the so-called modified Weibull distribution is highlighted and studied. It differs from the competition with a very flexible hazard rate function exhibiting increasing, decreasing, constant, upside-down bathtub and bathtub shapes. This panel of shapes remains rare and particularly desirable for modeling purposes. We provide the main mathematical properties of the special distribution, such as a tractable infinite series expansion of the probability density function, moments of several kinds (raw, incomplete, probability weighted\dots) with discussions on the skewness and kurtosis. The stochastic ordering structure and stress-strength parameter are also considered, as well as the basics of the order statistics. Then, an emphasis is put on the inferential features of the related model. In particular, the estimation of the model parameters is employed by the maximum likelihood method, with a simulation study to confirm the suitability of the approach. Three practical data sets are then analyzed. It is observed that the proposed model gives better fits than other well-known lifetime models derived from the Weibull model.A new extension of the Birnbaum-Saunders distributionhttps://www.zbmath.org/1483.600232022-05-16T20:40:13.078697Z"Lemonte, Artur J."https://www.zbmath.org/authors/?q=ai:lemonte.artur-joseSummary: In this paper, a new extension for the Birnbaum-Saunders distribution, which has been applied to the modeling of fatigue failure times and reliability studies, is introduced. The proposed model, called the Marshall-Olkin extended Birnbaum-Saunders distribution, arises based on the scheme introduced by \textit{A. W. Marshall} and \textit{I. Olkin} [Biometrika 84, No. 3, 641--652 (1997; Zbl 0888.62012)]. The maximum likelihood estimators and statistical inference for the new distribution parameters and influence diagnostic for the new distribution are presented. Finally, the proposed new distribution is applied to model three real data sets.The beta generalized logistic distributionhttps://www.zbmath.org/1483.600242022-05-16T20:40:13.078697Z"Morais, Alice L."https://www.zbmath.org/authors/?q=ai:morais.alice-lemos"Cordeiro, Gauss M."https://www.zbmath.org/authors/?q=ai:cordeiro.gauss-moutinho"Cysneiros, Audrey H. M. A."https://www.zbmath.org/authors/?q=ai:cysneiros.audrey-helen-m-aSummary: For the first time, a four-parameter beta generalized logistic distribution is obtained by compounding the beta and generalized logistic distributions. The new model extends some well-known distributions and its shape is quite flexible, specially the skewness and the tail weights, due to the extra shape parameters. We obtain general expansions for the moment generating and quantile functions. The estimation of the parameters is investigated by maximum likelihood. An application to a real data set is given to show the flexibility and potentiality of our distribution.Zero truncated negative binomial weighted Weibull distribution and its applicationhttps://www.zbmath.org/1483.600252022-05-16T20:40:13.078697Z"Sitho, Surang"https://www.zbmath.org/authors/?q=ai:sitho.surang"Denthet, Sunthree"https://www.zbmath.org/authors/?q=ai:denthet.sunthree"Nadeem, Hira"https://www.zbmath.org/authors/?q=ai:nadeem.hiraSummary: In this article, we propose two new discrete distributions, the negative binomial-weighted Weibull and the zero truncated negative binomial-weighted Weibull distributions. Some statistical properties of the proposed distributions are presented. The parameter estimates for the proposed distributions have been derived by the maximum likelihood estimation. The applications of the proposed distributions to real data sets in order to compare the performance with other distributions. The criteria for selecting the best fitted distribution are test statistics, i.e., the chi-square test and the Kolmogorov-Smirnov (K-S) test for discrete distributions.DS normal distribution: properties and applicationshttps://www.zbmath.org/1483.600262022-05-16T20:40:13.078697Z"Sulewski, P."https://www.zbmath.org/authors/?q=ai:sulewski.piotrSummary: The main aim of this paper is to introduce a new flexible distribution which generalizes the normal distribution. Some properties of the introduced distribution such as PDF, CDF, hazard function, quantiles, moments and generator are derived. Overdispersion, underdispersion and equidispersion are analyzed. The unknown parameters of the distribution are estimated by the maximum likelihood method. Illustrative examples of applicability and flexibility of the distribution in question are given. R software codes are presented in Appendix A.A family of probability distributions consistent with the DOZZ formula: towards a conjecture for the law of 2D GMChttps://www.zbmath.org/1483.600272022-05-16T20:40:13.078697Z"Ostrovsky, Dmitry"https://www.zbmath.org/authors/?q=ai:ostrovskii.dmitrii-mSummary: A three-parameter family of probability distributions is constructed such that its Mellin transform is defined over the same domain as the 2D GMC on the Riemann sphere with three insertion points \((\alpha_1,\alpha_2,\alpha_3)\) and satisfies the DOZZ formula in the sense of \textit{A. Kupiainen} et al. [Ann. Math. (2) 191, No. 1, 81--166 (2020; Zbl 1432.81055)]. The probability distributions in the family are defined as products of independent Fyodorov-Bouchaud and powers of Barnes beta distributions of types \((2, 1)\) and \((2, 2)\). In the special case of \(\alpha_1+\alpha_2+\alpha_3=2Q\) the constructed probability distribution is shown to be consistent with the known small deviation asymptotic of the 2D GMC laws with everywhere-positive curvature.Marshall-Olkin Esscher transformed Laplace distribution and processeshttps://www.zbmath.org/1483.600282022-05-16T20:40:13.078697Z"George, Dais"https://www.zbmath.org/authors/?q=ai:george.dais"George, Sebastian"https://www.zbmath.org/authors/?q=ai:george.sebastianSummary: In this article we consider a class of asymmetric distributions which belongs to one parameter regular exponential family. The Marshall-Olkin version of this family is also considered. Various properties are examined. Applications of these models in time series analysis are discussed. We also consider an application of Marshall-Olkin Esscher transformed Laplace distribution in financial modeling. A comparative study shows that Marshall-Olkin Esscher transformed Laplace distribution is a better fit to our data compared to asymmetric Laplace and Esscher transformed Laplace distributions.Sufficient conditions for some transform orders based on the quantile density ratiohttps://www.zbmath.org/1483.600292022-05-16T20:40:13.078697Z"Arriaza, Antonio"https://www.zbmath.org/authors/?q=ai:arriaza.antonio"Belzunce, Félix"https://www.zbmath.org/authors/?q=ai:belzunce.felix"Martínez-Riquelme, Carolina"https://www.zbmath.org/authors/?q=ai:martinez-riquelme.carolinaLet \(X\) and \(Y\) be two non-negative random variables with distribution functions \(F\) and \(G\) and density functions \(f\) and \(g\), respectively. It is said that \(X\) is smaller than \(Y\) in the \textit{convex transform order} if the \textit{quantile densities ratio} \(\displaystyle \frac{f\left(F^{-1}(p)\right)}{g\left(G^{-1}(p)\right)}\) is increasing in \(p\in(0,1)\), where \(F^{-1}\) and \(G^{-1}\) are the generalized inverses of \(F\) and \(G\), respectively. The paper studies the situations where the convex transform order does not hold but some other weaker transform orders are satisfied. Under the assumption that there exists a point \(p^*\in(0,1)\) such that the quantile densities ratio is increasing over \((0,p^*)\) and decreasing over \((p^*,1)\) (therefore the quantile densities ratio is unimodal), necessary and sufficient conditions for the \textit{star-shaped} ordering and for the \textit{expected proportional shortfall ordering} are proved. Also, assuming that the quantile densities ratio has a finite number of modes, the authors provide necessary and sufficient conditions for the \textit{qmit ordering} and for the \textit{dmrl ordering}. The results about the comparison of the non-negative random variables in various transform orders are then summarized and commented. In addition, applications in reliability are provided. The theoretical results are illustrated by numerical examples.
Reviewer: Eugen Paltanea (Braşov)Gaussian correlation inequality [after Thomas Royen]https://www.zbmath.org/1483.600302022-05-16T20:40:13.078697Z"Barthe, Franck"https://www.zbmath.org/authors/?q=ai:barthe.franckSummary: The Gaussian correlation conjecture predicts that for every centred Gaussian measure and any couple of convex origin-symmetric sets, the measure of their intersection is not less than the product of their individual measures. It was proved in two dimensions in the seventies, and despite its popularity and simple formulation, it has resisted until 2014. The proof of T. Royen uses, in an ingenious way, multivariate gamma distributions.
For the entire collection see [Zbl 1416.00029].Marginal and dependence uncertainty: bounds, optimal transport, and sharpnesshttps://www.zbmath.org/1483.600312022-05-16T20:40:13.078697Z"Bartl, Daniel"https://www.zbmath.org/authors/?q=ai:bartl.daniel"Kupper, Michael"https://www.zbmath.org/authors/?q=ai:kupper.michael"Lux, Thibaut"https://www.zbmath.org/authors/?q=ai:lux.thibaut"Papapantoleon, Antonis"https://www.zbmath.org/authors/?q=ai:papapantoleon.antonis"Eckstein, Stephan"https://www.zbmath.org/authors/?q=ai:eckstein.stephanProbability inequalities for sums of WUOD random variables and their applicationshttps://www.zbmath.org/1483.600322022-05-16T20:40:13.078697Z"Chen, Lamei"https://www.zbmath.org/authors/?q=ai:chen.lamei"Wang, Kaiyong"https://www.zbmath.org/authors/?q=ai:wang.kaiyong"Gao, Miaomiao"https://www.zbmath.org/authors/?q=ai:gao.miaomiao"Dong, Yilun"https://www.zbmath.org/authors/?q=ai:dong.yilunThe random variables \(\{ {X_i},i \geq 1\} \) are said to be widely upper orthant dependent (WUOD) if for each \(n = 1,2,...\) and all \({x_1},...,{x_n}\), there exists a finite real sequence \(\{ {g_U}(m),m \geq 1\} \) such that\(P({X_i} > {x_i},i = 1,2,...,n\} \le {g_U}(m)\prod\limits_{i = 1}^n {P({X_i} > {x_i})} \). They are said to be widely lower orthant dependent (WLOD) if there exists a finite real sequence \(\{ {g_L}(m),m \geq 1\} \) such that \(P({X_i} \leq {x_i},i = 1,2,...,n\} \leq {g_L}(m)\prod\limits_{i = 1}^n {P({X_i} \leq {x_i})} \). They are said to be widely orthant dependent (WOD) if they are both WUOD and WLOD.
The paper establishes inequalities for the tail probabilities of sums \({S_n} = {X_1} + ... + {X_n}\) with WUOD summands. As applications, the complete convergence of WOD random variables is investigated.
Reviewer: Oleg K. Zakusilo (Kyïv)Low correlation noise stability of symmetric setshttps://www.zbmath.org/1483.600332022-05-16T20:40:13.078697Z"Heilman, Steven"https://www.zbmath.org/authors/?q=ai:heilman.steven-mGaussian noise stability is a well-studied topic with connections to geometry of minimal surfaces [\textit{T. H. Colding} and \textit{W. P. Minicozzi II}, in: Surveys in geometric analysis and relativity. Dedicated to Richard Schoen in honor of his 60th birthday. Somerville, MA: International Press; Beijing: Higher Education Press. 73--143 (2011; Zbl 1261.53006)], hypercontractivity and invariance principles [\textit{E. Mossel} et al., Ann. Math. (2) 171, No. 1, 295--341 (2010; Zbl 1201.60031)], isoperimetric inequalities [\textit{D. M. Kane}, Comput. Complexity 23, No. 2, 151--175 (2014; Zbl 1314.68138)], sharp unique games hardness results in theoretical computer science [\textit{S. Khot} et al., SIAM J. Comput. 37, No. 1, 319--357 (2007; Zbl 1135.68019)], social choice theory, learning theory [\textit{A. R. Klivans} et al. ``Learning geometric concepts via Gaussian surface area'', in: IEEE 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS'08. Los Alamitos, CA: IEEE Computer Society. 541--550 (2008; \url{doi:10.1109/FOCS.2008.64})] and communication complexity [\textit{A. Chakrabarti} and \textit{O. Regev}, in: Proceedings of the 43rd annual ACM symposium on theory of computing, STOC '11. San Jose, CA, USA, June 6--8, 2011. New York, NY: Association for Computing Machinery (ACM). 51--60 (2011; Zbl 1288.90005)]. The author studies the Gaussian noise stability of subsets \(A\) of Euclidean space satisfying \(A =-A.\) It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)On the inequalities of Grüss-Čebyshev and Kantorovich: a probabilistic approachhttps://www.zbmath.org/1483.600342022-05-16T20:40:13.078697Z"Heinrich, Lothar"https://www.zbmath.org/authors/?q=ai:heinrich.lotharSummary: First we recall the original form of inequalities found by P. L. Čebyshev in 1882, G. Grüss in 1935 and V. L. Kantorovich in 1948. Then we formulate generalized versions of these inequalities in the language of probability theory which allows to prove them by simple probabilistic arguments. A further moment inequality of this type rounds off this note.Best lower bound on the probability of a binomial exceeding its expectationhttps://www.zbmath.org/1483.600352022-05-16T20:40:13.078697Z"Pinelis, Iosif"https://www.zbmath.org/authors/?q=ai:pinelis.iosif-froimovichLet \(X\sim Bin\, (n,p)\), and \(c = \ln(4/3)\). In this article, the author proves that \[\mbox{ if} \quad \frac{c}{n}\leq p < 1, \quad \mbox{ then } \quad P\left(X > E[X]\right) \geq \frac{1}{4}. \] The value of \(c\) is optimal. This result is a slight improvement of a result of \textit{S. Greenberg} and \textit{M. Mohri} [Stat. Probab. Lett. 86, 91--98 (2014; Zbl 1293.60024)].
``The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in learning theory and generalization bounds for unbounded loss functions.''
Reviewer: Italo Simonelli (Durham)Stochastic orders on two-dimensional space: application to cross entropyhttps://www.zbmath.org/1483.600362022-05-16T20:40:13.078697Z"Sbert, Mateu"https://www.zbmath.org/authors/?q=ai:sbert.mateu"Yoshida, Yuji"https://www.zbmath.org/authors/?q=ai:yoshida.yujiThe authors present the extension of the ordering between 1d probability mass functions (PMFs) to the ordering between 2d PMFs. They study the general case of non-independent random variables, and give some particular results for independent ones. They show also how the order between 2d PMFs submits the same order between the marginal probabilities. The paper introduces the concept of comonotonicity and shows its direct relationship with likelihood-ratio order. Section 2 contains a review of the first stochastic and likelihood-ratio between 1d PMFs. Section 3 introduces the stochastic order between 2d PMFs. Section 4 considers 1d and 2d comonotonic sequences and its relationship to the corresponding 1d or 2d likelihood-ratio order. Section 5 presents conclusions.
For the entire collection see [Zbl 1455.68026].
Reviewer: Oleg K. Zakusilo (Kyïv)Thin-shell theory for rotationally invariant random simpliceshttps://www.zbmath.org/1483.600372022-05-16T20:40:13.078697Z"Heiny, Johannes"https://www.zbmath.org/authors/?q=ai:heiny.johannes"Johnston, Samuel"https://www.zbmath.org/authors/?q=ai:johnston.samuel-g-g"Prochno, Joscha"https://www.zbmath.org/authors/?q=ai:prochno.joschaSummary: For fixed functions \(G,H : [0,\infty)\to [0,\infty)\), consider the rotationally invariant probability density on \(\mathbb{R}^n\) of the form
\[
\mu^n (\mathrm{d}s)=\frac{1}{Z_n} G(\| s\|_2)\mathrm{e}^{-nH(\| s\|_2)}\mathrm{d}s.
\]
We show that when \(n\) is large, the Euclidean norm \(\| Y^n\|_2\) of a random vector \(Y^n\) distributed according to \(\mu^n\) satisfies a thin-shell property, in that its distribution is highly likely to concentrate around a value \(s_0\) minimizing a certain variational problem. Moreover, we show that the fluctuations of this modulus away from \(s_0\) have the order \(1/\sqrt{n}\) and are approximately Gaussian when \(n\) is large.
We apply these observations to rotationally invariant random simplices: the simplex whose vertices consist of the origin as well as independent random vectors \(Y_1^n,\ldots,Y_p^n\) distributed according to \(\mu^n\), ultimately showing that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior. Our class of measures includes the Gaussian distribution, the beta distribution and the beta prime distribution on \(\mathbb{R}^n\), provided a generalizing and unifying setting for the objects considered in [\textit{J. Grote} et al., ALEA, Lat. Am. J. Probab. Math. Stat. 16, No. 1, 141--177 (2019; Zbl 1456.52006)].
Finally, the volumes of random simplices may be related to the determinants of random matrices, and we use our methods with this correspondence to show that if \(A^n\) is an \(n\times n\) random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants \(c_0, c_1\in (0,\infty)\) and an absolute constant \(C\in (0,\infty)\) such that
\[
\underset{s\in \mathbb{R}}{\sup} \left| \mathbb{P}\left[ \dfrac{\log \mathrm{det} (A^n) -\log (n-1)\! -c_0}{\sqrt{\frac{1}{2}\log n+c_1}} < s\right] -\int_{-\infty}^s \dfrac{\mathrm{e}^{-u^2 /2} \mathrm{d}u}{\sqrt{2\pi}} \right| < \dfrac{C}{\log^{3/2}n},
\]
sharpening the \(1/\log^{1/3+o(1)} n\) bound in [\textit{H. H. Nguyen} and \textit{V. Vu}, Ann. Probab. 42, No. 1, 146--167 (2014; Zbl 1299.60005)].The birth of random evolutionshttps://www.zbmath.org/1483.600382022-05-16T20:40:13.078697Z"Hersh, Reuben"https://www.zbmath.org/authors/?q=ai:hersh.reubenFrom the text: The theory of random evolutions was born in Albuquerque in the late 1960s, flourished and matured in the 1970s, sprouted a robust daughter in Kiev in the 1980s, and is today a tool or method, applicable in a variety of ``real-world'' ventures.Limit theorems for additive functionals of random walks in random sceneryhttps://www.zbmath.org/1483.600392022-05-16T20:40:13.078697Z"Pène, Françoise"https://www.zbmath.org/authors/?q=ai:pene.francoiseThe authors study the asymptotic behaviour of additive functionals of random walks in random scenery (RWRS). It is the process defined as follows: \[Z_n :=\sum_{k=0}^{n-1}\xi_{S_k}=\sum_{y\in \mathbb{Z}}\xi_yN_n(y),\] where \(S\) is a random walk, \(N_n(y)\) stands for the local time of \(S\) at position \(y\) before time \(n\). This process was first studied by Borodin, Kesten and Spitzer, and it describes the evolution of the total amount won until time \(n\) by a particle moving with respect to the random walk \(S\), starting with a null amount at time \(0\) and winning the amount \(\xi_l\)` at each time the particle visits the position \(l\in \mathbb{Z}\). This process is a natural example of (strongly) stationary process with long time dependence. The authors establish bounds for the moments of the local time of this process. These bounds combined with a previous moment convergence result (and an ergodicity result) imply the convergence in distribution of additive observables (with a normalization like \(n^{1/4}\)). When the sum of the observables is null, the previous limit vanishes and the convergence in the sense of moments (with a normalization like \(n^{1/8}\)) is established.
Reviewer: Yuliya S. Mishura (Kyïv)Poisson approximation with applications to stochastic geometryhttps://www.zbmath.org/1483.600402022-05-16T20:40:13.078697Z"Pianoforte, Federico"https://www.zbmath.org/authors/?q=ai:pianoforte.federico"Schulte, Matthias"https://www.zbmath.org/authors/?q=ai:schulte.matthiasSummary: This article compares the distributions of integer-valued random variables and Poisson random variables. It considers the total variation and the Wasserstein distance and provides, in particular, explicit bounds on the pointwise difference between the cumulative distribution functions. Special attention is dedicated to estimating the difference when the cumulative distribution functions are evaluated at 0. This permits to approximate the minimum (or maximum) of a collection of random variables by a suitable random variable in the Kolmogorov distance. The main theoretical results are obtained by combining the Chen-Stein method with size-bias coupling and a generalization of size-bias coupling for integer-valued random variables developed herein. A wide variety of applications are then discussed with a focus on stochastic geometry. In particular, transforms of the minimal circumscribed radius and the maximal inradius of Poisson-Voronoi tessellations as well as the minimal inter-point distance of the points of a Poisson process are considered and bounds for their Kolmogorov distances to extreme value distributions are derived.Persistence exponents in Markov chainshttps://www.zbmath.org/1483.600412022-05-16T20:40:13.078697Z"Aurzada, Frank"https://www.zbmath.org/authors/?q=ai:aurzada.frank"Mukherjee, Sumit"https://www.zbmath.org/authors/?q=ai:mukherjee.sumit"Zeitouni, Ofer"https://www.zbmath.org/authors/?q=ai:zeitouni.oferSummary: We prove the existence of the \textit{persistence exponent}
\[
\log\lambda :=\lim\limits_{n\to \infty}\frac{1}{n}\log{\mathbb{P}_{\mu}}({X_0}\in S,\dots,{X_n}\in S)
\]
for a class of time homogeneous Markov chains \(\{X_i\}_{i\ge 0}\) taking values in a Polish space, where \(S\) is a Borel measurable set and \(\mu\) is an initial distribution. Focusing on the case of \(\mathrm{AR}(p)\) and \(\mathrm{MA}(q)\) processes with \(p,q\in\mathbb{N}\) and continuous innovation distribution, we study the existence of \(\lambda\) and its continuity in the parameters of the AR and MA processes, respectively, for \(S=\mathbb{R}_{\ge 0}\). For AR processes with log-concave innovation distribution, we prove the strict monotonicity of \(\lambda\). Finally, we compute new explicit exponents in several concrete examples.Pathwise asymptotics for Volterra type stochastic volatility modelshttps://www.zbmath.org/1483.600422022-05-16T20:40:13.078697Z"Cellupica, Miriana"https://www.zbmath.org/authors/?q=ai:cellupica.miriana"Pacchiarotti, Barbara"https://www.zbmath.org/authors/?q=ai:pacchiarotti.barbaraSummary: We study stochastic volatility models in which the volatility process is a positive continuous function of a continuous Volterra stochastic process. We state some pathwise large deviation principles for the scaled log-price.Large deviations for the right-most position of a last progeny modified branching random walkhttps://www.zbmath.org/1483.600432022-05-16T20:40:13.078697Z"Ghosh, Partha Pratim"https://www.zbmath.org/authors/?q=ai:ghosh.partha-pratim|ghosh.partha-pratim.2|ghosh.partha-pratim.1Summary: In this work, we consider a modification of the usual \textit{Branching Random Walk (BRW)}, where we give certain independent and identically distributed (i.i.d.) displacements to all the particles at the \(n\)-th generation, which may be different from the driving increment distribution. This model was first introduced by \textit{A. Bandyopadhyay} and \textit{P. P. Ghosh} [``Right-most position of a last progeny modified branching random walk'', Preprint, \url{arXiv:2106.02880}] and they termed it as \textit{Last Progeny Modified Branching Random Walk (LPM-BRW)}. Under very minimal assumptions, we derive the \textit{large deviation principle (LDP)} for the right-most position of a particle in generation \(n\). As a byproduct, we also complete the LDP for the classical model, which complements the earlier work by \textit{N. Gantert} and \textit{T. Höfelsauer} [Electron. Commun. Probab. 23, Paper No. 34, 12 p. (2018; Zbl 1394.60019)].Moderate deviations for extreme eigenvalues of real-valued sample covariance matriceshttps://www.zbmath.org/1483.600442022-05-16T20:40:13.078697Z"Jiang, Hui"https://www.zbmath.org/authors/?q=ai:jiang.hui"Wang, Shaochen"https://www.zbmath.org/authors/?q=ai:wang.shaochen"Zhou, Wang"https://www.zbmath.org/authors/?q=ai:zhou.wangSummary: Consider the sample covariance matrices of form \(W=n^{-1}CC^{\top}\), where \(C\) is a \(k\times n\) matrix with real-valued, independent and identically distributed (i.i.d.) mean zero entries. When the squares of the i.i.d. entries have finite exponential moments, the moderate deviations for the extreme eigenvalues of \(W\) are investigated as \(n\rightarrow \infty\) and either \(k\) is fixed or \(k\rightarrow\infty\) with some suitable growth conditions. The moderate deviation rate function reveals that the right (left) tail of \(\lambda_{\max}\) is more like Gaussian rather than the Tracy-Widom type distribution when \(k\) goes to infinity slowly.Asymptotic behaviors for correlated Bernoulli modelhttps://www.zbmath.org/1483.600452022-05-16T20:40:13.078697Z"Miao, Yu"https://www.zbmath.org/authors/?q=ai:miao.yu"Ma, Huanhuan"https://www.zbmath.org/authors/?q=ai:ma.huanhuan"Yang, Qinglong"https://www.zbmath.org/authors/?q=ai:yang.qinglongSummary: We consider a class of correlated Bernoulli variables, which have the following form: for some \(0<p<1\),
\[
P(X_{j+1}=1\mid\mathcal{F}_j)=(1-\theta_j)p+\theta_jS_j/j,
\]
where \(0\leq\theta_j\leq 1\), \(S_n=\sum_{j=1}^nX_j\) and \(\mathcal{F}_n=\sigma\{X_1,\dots,X_n\}\). The aim of this paper is to establish the strong law of large numbers which extend some known results, and prove the moderate deviation principle for the correlated Bernoulli model.Some large deviation asymptotics in small noise filtering problemshttps://www.zbmath.org/1483.600462022-05-16T20:40:13.078697Z"Reddy, Anugu Sumith"https://www.zbmath.org/authors/?q=ai:reddy.anugu-sumith"Budhiraja, Amarjit"https://www.zbmath.org/authors/?q=ai:budhiraja.amarjit-s"Apte, Amit"https://www.zbmath.org/authors/?q=ai:apte.amitComplete moment convergence for weighted sums of extended negatively dependent random variableshttps://www.zbmath.org/1483.600472022-05-16T20:40:13.078697Z"Ge, Meimei"https://www.zbmath.org/authors/?q=ai:ge.meimei"Deng, Xin"https://www.zbmath.org/authors/?q=ai:deng.xinThe paper generalizes the following result of \textit{L. E. Baum} and \textit{M. Katz} [Trans. Am. Math. Soc. 120, 108--123 (1965; Zbl 0142.14802)] on convergence rates in the law of large numbers: Let \(p > 1/\alpha \) and\({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} < \alpha \le 1\). Let \(\{ {X_n},n \ge 1\} \) be a sequence of independent and identically distributed random variables with\(E{X_1} = 0\). If \(E{\left| {{X_1}} \right|^p} < \infty \), then for \(\varepsilon > 0\),\(\sum\limits_{n = 1}^\infty {{n^{\alpha p - 2}}P\left( {\left| {{X_i}} \right| > \varepsilon {n^\alpha }} \right)} < \infty \). The authors improve and extend it to weighted sums of extended negatively dependent (END) random variables. Their results mainly make the following three improvements. (i) They are established from complete convergence for independent random variables to complete moment convergence for END random variables; (ii) They are established from non-weighted sums to weighted sums, and the condition on weights is very mild; (iii) They are established from \({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} < \alpha \le 1\) to \({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} < \alpha \), and from \(\alpha p > 1\) to \(\alpha p \ge 1\).
Reviewer: Oleg K. Zakusilo (Kyïv)On the complete moment convergence of weighted sums of \(\rho^*\)-mixing random variableshttps://www.zbmath.org/1483.600482022-05-16T20:40:13.078697Z"Ge, Meimei"https://www.zbmath.org/authors/?q=ai:ge.meimei"Lv, Wenhua"https://www.zbmath.org/authors/?q=ai:lv.wenhua"Wu, Yongfeng"https://www.zbmath.org/authors/?q=ai:wu.yongfengThe paper continues investigations of \({\rho ^ * }\)-mixing sequences of random variables. The authors study the complete moment convergence of the weighted sum of \({\rho ^ * }\) -mixing sequences, which are stochastically dominated by a random variable \(X\).
Reviewer: Oleg K. Zakusilo (Kyïv)Limit theorems for discounted convergent perpetuitieshttps://www.zbmath.org/1483.600492022-05-16T20:40:13.078697Z"Iksanov, Alexander"https://www.zbmath.org/authors/?q=ai:iksanov.aleksander-m"Nikitin, Anatolii"https://www.zbmath.org/authors/?q=ai:nikitin.anatolii-v"Samoilenko, Igor"https://www.zbmath.org/authors/?q=ai:samoilenko.igor-vSummary: Let \((\xi_1,\eta_1), (\xi_2,\eta_2),\ldots\) be independent identically distributed \(\mathbb{R}^2\)-valued random vectors. We prove a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm for the convergent perpetuities \(\sum_{k\geq 0} b^{\xi_1 +\ldots +\xi_k} \eta_{k+1}\) as \(b\to 1-\). Under the standard actuarial interpretation, these results correspond to the situation when the actuarial market is close to the customer-friendly scenario of no risk.On the complete convergence for weighted sums of extended negatively dependent random variableshttps://www.zbmath.org/1483.600502022-05-16T20:40:13.078697Z"Wu, Yongfeng"https://www.zbmath.org/authors/?q=ai:wu.yongfeng"Zhai, Mingqing"https://www.zbmath.org/authors/?q=ai:zhai.mingqing"Peng, Jiangyan"https://www.zbmath.org/authors/?q=ai:peng.jiangyanThe random variables \(\{ {X_i},i \ge 1\} \) are said to be extended negatively dependent (END) random variables if for each \(n = 1,2,\ldots\) and all \({x_1},...,{x_n}\), there exists a constant \(M > 0\) such that both \(P({X_i} \le {x_i},i = 1,2,\ldots,n\} \le M\prod\limits_{i = 1}^n {P({X_i} \le {x_i})} \) and \(P({X_i} > {x_i},i = 1,2,...,n\} \le M\prod\limits_{i = 1}^n {P({X_i} > {x_i})} \) hold. They are said to be negatively orthant dependent (NOD), if\(M = 1\). The paper investigates the complete convergence for weighted sums of END random variables. It extends and improves results obtained earlier by considering END instead of NOD, the maximal partial sums instead of the common partial sums, obtaining some stronger conclusions under the same or weaker conditions
Reviewer: Oleg K. Zakusilo (Kyïv)Functional limit theorems for power series with rapid decay of moving averages of Hermite processeshttps://www.zbmath.org/1483.600512022-05-16T20:40:13.078697Z"Gehringer, Johann Rudolf"https://www.zbmath.org/authors/?q=ai:gehringer.johann-rudolfA path formula for the sock sorting problemhttps://www.zbmath.org/1483.600522022-05-16T20:40:13.078697Z"Korbel, S."https://www.zbmath.org/authors/?q=ai:korbel.s"Mörters, P."https://www.zbmath.org/authors/?q=ai:morters.peterVariable speed symmetric random walk driven by the simple symmetric exclusion processhttps://www.zbmath.org/1483.600532022-05-16T20:40:13.078697Z"Menezes, Otávio"https://www.zbmath.org/authors/?q=ai:menezes.otavio"Peterson, Jonathon"https://www.zbmath.org/authors/?q=ai:peterson.jonathon"Xie, Yongjia"https://www.zbmath.org/authors/?q=ai:xie.yongjiaSummary: We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. This model can be viewed as a special case of the random walk in a balanced random environment, for which the weak quenched limit is constructed as a function of the invariant measure of the environment viewed from the walk. We bypass the need to show the existence of this invariant measure. Instead, we find the limit of the quadratic variation of the walk and give an explicit formula for it.Exponential concentration for zeroes of stationary Gaussian processeshttps://www.zbmath.org/1483.600542022-05-16T20:40:13.078697Z"Basu, Riddhipratim"https://www.zbmath.org/authors/?q=ai:basu.riddhipratim"Dembo, Amir"https://www.zbmath.org/authors/?q=ai:dembo.amir"Feldheim, Naomi"https://www.zbmath.org/authors/?q=ai:feldheim.naomi-dvora"Zeitouni, Ofer"https://www.zbmath.org/authors/?q=ai:zeitouni.oferSummary: We show that for any centered stationary Gaussian process of absolutely integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in \([0,T]\) is within \(\eta T\) of its mean value, up to an exponentially small in \(T\) probability.Random harmonic processes with new propertieshttps://www.zbmath.org/1483.600552022-05-16T20:40:13.078697Z"Karachanskaya, Elena"https://www.zbmath.org/authors/?q=ai:karachanskaya.elena-vSummary: In this paper, we introduce a new general model for random signals and adjoining harmonic processes. This model is stochastic hierarchically correlated series (SHCS). The sufficient conditions for both the wide-sense stationary property and the mean ergodic property for random harmonic (trigonometric) processes are set. Moreover, we describe a class of wide-sense stationary and mean ergodic random harmonic processes function with nonuniformly random phase.
For the entire collection see [Zbl 1470.46003].Packing dimension of space-time anisotropic Gaussian random fieldshttps://www.zbmath.org/1483.600562022-05-16T20:40:13.078697Z"Chen, Zhen Long"https://www.zbmath.org/authors/?q=ai:chen.zhenlong"Wang, Jun"https://www.zbmath.org/authors/?q=ai:wang.jun.14|wang.jun.9|wang.jun|wang.jun.11|wang.jun.3|wang.jun.13|wang.jun.12|wang.jun.2|wang.jun.6|wang.jun.1"Wu, Dong Sheng"https://www.zbmath.org/authors/?q=ai:wu.dongshengSummary: Let \(X= \{ X(t) \in \mathbb{R}^d, t \in \mathbb{R}^N\}\) be a centered space-time anisotropic Gaussian random field whose components are independent and satisfy some mild conditions. We study the packing dimension of range \(X(E)\) under the anisotropic (time variable) metric space \((\mathbb{R}^N, \rho)\) and (space variable) metric space \((\mathbb{R}^d, \tau)\), where \(E \subset \mathbb{R}^N\) is a Borel set. Our results generalize the corresponding results of \textit{A. Estrade} et al. [Commun. Stoch. Anal. 5, No. 1, 41--64 (2011; Zbl 1331.60057)] for time-anisotropic Gaussian random fields to space-time anisotropic Gaussian fields.The probability of first reaching a desired level by a random process on a given intervalhttps://www.zbmath.org/1483.600572022-05-16T20:40:13.078697Z"Semakov, S. L."https://www.zbmath.org/authors/?q=ai:semakov.s-l"Semakov, I. S."https://www.zbmath.org/authors/?q=ai:semakov.i-sSummary: This paper estimates the probability of an event that a continuous random process first reaches a desired level on a given interval of the independent variable. The results are specified for Gaussian processes. An example of numerical bounds is given.On temporal regularity of stochastic convolutions in \(2\)-smooth Banach spaceshttps://www.zbmath.org/1483.600582022-05-16T20:40:13.078697Z"Ondreját, Martin"https://www.zbmath.org/authors/?q=ai:ondrejat.martin"Veraar, Mark"https://www.zbmath.org/authors/?q=ai:veraar.mark-cThe class of 2-smooth Banach spaces plays an important role in stochastic analysis in infinite dimensions. To overcome the fact that there is no smallest Hölder space or Besov space to which Brownian path belongs to almost surely, in this paper more general Hölder spaces are considered, called modulus Hölder spaces. In this paper some known results are generalized not only to infinite-dimensional stochastic integrals but also to stochastic convolutions in 2-smooth Banach spaces, and, consequently, it is shown that path of mild solutions to parabolic stochastic differential equations in any 2-smooth Banach space \(X\) have path in the Besov-Orlicz space \(B^{1/2}_{\Phi_2,\infty}(0,T;X)\) almost surely, where \(\Phi_2(x)=\exp(x^2)-1\). To do this, after the presentation of the topic and the main results in the first section, in the second section Orlicz spaces, Besov-Orlicz spaces, and embeddings to Hölder spaces, besides other necessary things are presented. In the third section temporal regularity of stochastic integral is studied, the main result from here (Theorem 3.2) providing the optimal path regularity properties and norm estimates. Temporal regularity of deterministic convolutions is studied in the fourth section and a maximal regularity in the scale of Besov-Orlicz is discussed. In the last section of the paper, the main theorem (Theorem 1.1) is proved applying the results obtained in the previous two sections.
Reviewer: Ilie Valuşescu (Bucureşti)Multifractal processes: definition, properties and new exampleshttps://www.zbmath.org/1483.600592022-05-16T20:40:13.078697Z"Grahovac, Danijel"https://www.zbmath.org/authors/?q=ai:grahovac.danijelSummary: We investigate stochastic processes possessing scale invariance properties which we refer to as multifractal processes. The examples of such processes known so far do not go much beyond the original cascade construction of Mandelbrot. We provide a new definition of the multifractal process by generalizing the definition of the self-similar process. We establish general properties of these processes and show how existing examples fit into our setting. Finally, we define a new class of examples inspired by the idea of Lamperti transformation. Namely, for any pair of infinitely divisible distribution and a stationary process one can construct a multifractal process.Fractional Lévy stable motion: finite difference iterative forecasting modelhttps://www.zbmath.org/1483.600602022-05-16T20:40:13.078697Z"Liu, He"https://www.zbmath.org/authors/?q=ai:liu.he"Song, Wanqing"https://www.zbmath.org/authors/?q=ai:song.wanqing"Li, Ming"https://www.zbmath.org/authors/?q=ai:li.ming"Kudreyko, Aleksey"https://www.zbmath.org/authors/?q=ai:kudreyko.aleksey"Zio, Enrico"https://www.zbmath.org/authors/?q=ai:zio.enricoSummary: In this study we use the fractional Lévy stable motion (fLsm) to establish a finite iterative forecasting model with Long Range Dependent (LRD) characteristics. The LRD forecasting model considers the influence of current and past trends in stochastic sequences on future trends. We find that the discussed model can accurately forecast the trends of stochastic sequences. This fact enables us to introduce the fLsm as the fractional-order model of Lévy stable motion. Self-similarity and LRD characteristics of the flsm model is introduced by using the relationship between self-similar index and the characteristic index. Thus, the order Stochastic Differential Equation (FSDE) which describes the fLsm can be obtained. The parameters of the FSDE were estimated by using a novel characteristic function method. The forecasting model with the LRD characteristics was obtained by discretization of FSDE. The Monte Carlo method was applied to demonstrate the feasibility of the forecasting model. The power load forecasting history data demonstrates the advantages of our model.Spectral analysis for some multifractional Gaussian processeshttps://www.zbmath.org/1483.600612022-05-16T20:40:13.078697Z"Karol, A. I."https://www.zbmath.org/authors/?q=ai:karol.andrei-i"Nazarov, A. I."https://www.zbmath.org/authors/?q=ai:nazarov.alexander-iSummary: We study the small ball asymptotics problem in \(L_2\) for two generalizations of the fractional Brownian motion with variable Hurst parameter. To this end, we perform a careful analysis of the singular value asymptotics for associated integral operators.Constrained maximum variance stopping for a finite horizon increasing random walkhttps://www.zbmath.org/1483.600622022-05-16T20:40:13.078697Z"Fontem, Belleh"https://www.zbmath.org/authors/?q=ai:fontem.belleh-aSummary: We derive the optimal solution for the problem of choosing a non-anticipative decision rule to maximize the stopping variance of a finite horizon, increasing random walk subject to a distributional constraint, as well as an explicit upper limit on the variance of the walk's stopping state. Problems of this caliber arise as subproblems for risk-constrained versions of standard stopping problems in areas including, for instance, market entry decision-making. A numerical example verifies the main result.On martingale methods and some Besicovitch type setshttps://www.zbmath.org/1483.600632022-05-16T20:40:13.078697Z"Paszkiewicz, Adam"https://www.zbmath.org/authors/?q=ai:paszkiewicz.adam"Prusinowski, Damian"https://www.zbmath.org/authors/?q=ai:prusinowski.damianThis paper provides the existence of a \textit{Besicovitch} set on \(\mathbb{R}^2\) through a martingale convergence argument. A \textit{Besicovitch} set in \(\mathbb{R}^2\) is a Borel subset of \(\mathbb{R}^2\) of Lebesque measure 0 which contains a copy of the unit interval \(I\) in every direction. The main theorem that is proved in this paper provides the existence of a subset of \(I\times I\) which is a union of segments in a range of directions having total Lebesque measure equal to 0. A Besicovitch set is then constructed out of this. The proof of the main theorem relies on a geometric construction which gives rise to a supermartingale with respect to an appropriately defined filtration. Here the random variable is defined through indicator functions on subsets of \(I\times I\). The filtration comes from a sequence of refining partitions on \(I\times I\). The existence of the set which is given in the main theorem is deduced from the martingale convergence theorem.
Reviewer: Nikolaos Fountoulakis (Birmingham)Single jump filtrations: preservation of the local martingale property with respect to the filtration generated by the local martingalehttps://www.zbmath.org/1483.600642022-05-16T20:40:13.078697Z"Gushchin, Alexander A."https://www.zbmath.org/authors/?q=ai:gushchin.alexander-a"Zhunussova, Assylliya K."https://www.zbmath.org/authors/?q=ai:zhunussova.assylliya-kSummary: Let \(M\) be a local martingale with respect to a so-called single jump filtration \(\mathbb{F}=\mathbb{F}(\gamma ,\mathcal{F})\) generated by a random time \(\gamma\) on a probability space \((\varOmega ,\mathcal{F},\mathsf P)\). It was recently mentioned by \textit{M. Herdegen} and \textit{S. Herrmann} [Stochastic Processes Appl. 126, No. 2, 337--359 (2016; Zbl 1329.60113)] that \(M\) is also a local martingale with respect to the filtration \(\mathbb{H}=\mathbb{F}^M\) that it generates if \(\mathcal{F}\) is the smallest \(\sigma \)-field with respect to which \(\gamma\) is measurable. We provide an example of a local martingale with respect to a general single jump filtration which is not a local martingale with respect to \(\mathbb{H} \). Then, we find necessary and sufficient condition for preserving the local martingale property with respect to \(\mathbb{H} \). The main idea of our constructions and the proofs is that \(\mathbb{H}\) is also a single jump filtration generated, in general, by other random time and \(\sigma \)-field. Finally, we prove that every \(\sigma \)-martingale in considered models is still a \(\sigma \)-martingale with respect to the filtration that it generates.
For the entire collection see [Zbl 1470.46003].Scaling exponents of step-reinforced random walkshttps://www.zbmath.org/1483.600652022-05-16T20:40:13.078697Z"Bertoin, Jean"https://www.zbmath.org/authors/?q=ai:bertoin.jeanLet \((\varepsilon_n)_{n\geq 1}\) be a \(0-1\)-valued deterministic sequence with \(\varepsilon_1=1\); \(X_1\), \(X_2,\ldots\) independent identically distributed real-valued random variables; \(U(2)\), \(U(3),\ldots\) independent random variables such that \(U(n)\) has a discrete uniform distribution on \(\{1,\ldots, n-1\}\), the two sequences being independent. Let \((S_n)_{n\geq 1}\) be the standard random walk with jumps \(X_k\) and assume that the distribution of \(X_1\) belongs to the normal domain of attraction of an \(\alpha\)-stable distribution, \(\nu_\alpha\) say, for \(\alpha\in (0,2]\). The latter means that \(\lim_{n\to\infty}n^{-1/\alpha}S_n=Y_\alpha\) in distribution, where \(Y_\alpha\) is a random variable with distribution \(\nu_\alpha\). The parameter \(\alpha\) is known as scaling exponent. Further, consider a perturbed random walk \((\hat S_n)_{n\geq 1}\) that the author calls a step-reinforced random walk defined by \(\hat S_n:=\hat X_1+\ldots+\hat X_n\) for \(n\in\mathbb{N}\), where \(\hat X_n:=X_{\varepsilon_1+\ldots+\varepsilon_n}\) if \(\varepsilon_n=1\) and \(\hat X_n:=\hat X_{U(n)}\) if \(\varepsilon_n=0\).
The purpose of the paper is to find out how the reinforcement affects the distributional growth of partial sums. To exemplify the author's findings, assume that \(\sum_{n\geq 1}n^{-2}|\varepsilon_1+\ldots+\varepsilon_n-qn|<\infty\) for some \(q\in (0,1)\) and that \(q>1-1/\alpha\) when \(\alpha>1\). Then the scaling exponents of \(S\) and \(\hat S\) coincide and are equal to \(\alpha\). If \(q<1-1/\alpha\), then the scaling exponent of \(\hat S\) is \((1-q)^{-1}<\alpha\). The case in which \(\varepsilon_1+\cdots+\varepsilon_n\) exhibits a sublinear growth is also analyzed. The main results of the paper are accompanied with appealing intuitive explanations. The paper finishes with a number of interesting remarks which particularly suggest several directions for future work.
Reviewer: Alexander Iksanov (Kiev)Quenched invariance principle for random walks on dynamically averaging random conductanceshttps://www.zbmath.org/1483.600662022-05-16T20:40:13.078697Z"Bethuelsen, Stein Andreas"https://www.zbmath.org/authors/?q=ai:bethuelsen.stein-andreas"Hirsch, Christian"https://www.zbmath.org/authors/?q=ai:hirsch.christian"Mönch, Christian"https://www.zbmath.org/authors/?q=ai:monch.christianSummary: We prove a quenched invariance principle for continuous-time random walks in a dynamically averaging environment on \(\mathbb{Z}\). In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk.Switching interacting particle systems: scaling limits, uphill diffusion and boundary layerhttps://www.zbmath.org/1483.600672022-05-16T20:40:13.078697Z"Floreani, Simone"https://www.zbmath.org/authors/?q=ai:floreani.simone"Giardinà, Cristian"https://www.zbmath.org/authors/?q=ai:giardina.cristian"den Hollander, Frank"https://www.zbmath.org/authors/?q=ai:den-hollander.frank"Nandan, Shubhamoy"https://www.zbmath.org/authors/?q=ai:nandan.shubhamoy"Redig, Frank"https://www.zbmath.org/authors/?q=ai:redig.frankSummary: This paper considers three classes of interacting particle systems on \(\mathbb{Z}\): independent random walks, the exclusion process, and the inclusion process. Particles are allowed to switch their jump rate (the rate identifies the \textit{type} of particle) between 1 (\textit{fast particles}) and \(\epsilon\in[0,1]\) (\textit{slow particles}). The switch between the two jump rates happens at rate \(\gamma\in(0,\infty)\). In the exclusion process, the interaction is such that each site can be occupied by at most one particle of each type. In the inclusion process, the interaction takes places between particles of the same type at different sites and between particles of different type at the same site. We derive the macroscopic limit equations for the three systems, obtained after scaling space by \(N^{-1}\), time by \(N^2\), the switching rate by \(N^{-2}\), and letting \(N\rightarrow \infty\). The limit equations for the macroscopic densities associated to the fast and slow particles is the well-studied double diffusivity model. This system of reaction-diffusion equations was introduced to model polycrystal diffusion and dislocation pipe diffusion, with the goal to overcome the limitations imposed by Fick's law. In order to investigate the microscopic out-of-equilibrium properties, we analyse the system on \([N]=\{1,\dots,N\}\), adding boundary reservoirs at sites 1 and \(N\) of fast and slow particles, respectively. Inside \([N]\) particles move as before, but now particles are injected and absorbed at sites 1 and \(N\) with prescribed rates that depend on the particle type. We compute the steady-state density profile and the steady-state current. It turns out that uphill diffusion is possible, i.e., the total flow can be in the direction of increasing total density. This phenomenon, which cannot occur in a single-type particle system, is a violation of Fick's law made possible by the switching between types. We rescale the microscopic steady-state density profile and steady-state current and obtain the steady-state solution of a boundary-value problem for the double diffusivity model.Least gradient functions in metric random walk spaceshttps://www.zbmath.org/1483.600682022-05-16T20:40:13.078697Z"Górny, Wojciech"https://www.zbmath.org/authors/?q=ai:gorny.wojciech"Mazón, José M."https://www.zbmath.org/authors/?q=ai:mazon-ruiz.jose-mThe authors study least gradient functions in the general setting of metric random walk spaces. A metric random walk space \([X,d,m]\) is a Polish metric space \((X,d)\) with a random walk \(m\), i.e., a family of probability measures \((m_x)_{x\in X}\) on the Borel \(\sigma\)-algebra of \(X\) describing the distribution of jumps from \(x\). Assuming the existence and uniqueness of an invariant and reversible measure \(\nu\) with \(\nu(X)<\infty\) and validity of the following \emph{\(p\)-Poincaré Inequality} in some \(\Omega\) with \(0<\nu(\Omega)<\nu(X)\) for all \(p\geq1\):
\begin{align*}
&\text{ there exists }\,\lambda>0\,:\quad \text{ for all }\,\, u\in L^p(\Omega,\nu),\quad\text{ for all }\,\,\psi\in L^p(\partial_m\Omega),\\
&\lambda\int_{\Omega}|u(x)|^pd\nu(x)\leq \int_\Omega\int_{\Omega_m}|u_\psi(y)-u(x)|^pdm_x(y)d\nu(x)+\int_{\partial_m\Omega}|\psi(x)|^pd\nu(y),
\end{align*}
where \(\partial_m\Omega:=\{x\in X\,:\,m_x(\Omega)>0 \}\) is \emph{\(m\)-boundary} of \(\Omega\), \(\Omega_m:=\Omega\cup\partial_m\Omega\), \(u_\psi(x):=u(x)\) for \(x\in\Omega\) and \(u_\psi(x):=\psi(x)\) for \(x\in\partial_m\Omega\), the authors prove the following theorem:
Consider the space of functions of \(m\)-bounded variation \[BV_m(X):=\left\{f\,:\,X\to\mathbb{R}\,\,\nu\text{-measurable}\,\,:\,\,\int_X\int_X|f(y)-f(x)|dm_x(y)d\nu(x)<\infty \right\}.\] Let \(\psi\in L^\infty(X\setminus\Omega)\) and \(u\in BV_m(X)\) such that \(u=\psi\) \(\,\nu\)-a.s. on \(X\setminus\Omega\). Then the following statements are equivalent:
\begin{itemize}
\item[(i)] \(u\big|_\Omega\) is a minimizer of the energy functional \(\mathcal{J}_\psi\), where
\begin{align*}
\mathcal{J}_\psi(u):=TV_m(u_\psi):=\frac12\int_{\Omega_m}\int_{\Omega_m}|u_\psi(y)-u_\psi(x)|dm_x(y)d\nu(x);
\end{align*}
\item[(ii)] \(u\big|_\Omega\) is a solution to the nonlocal \(1\)-Laplace problem with Dirichlet boundary condition \(\psi\)
\begin{align*}
\left\{ \begin{array}{ll} -\Delta_1^m u(x)=0, & x\in\Omega,\\
u(x)=\psi(x), & x\in\partial_m\Omega, \end{array} \right.
\end{align*}
where \(\Delta_1^m u(x):=\int_{\Omega_m}\frac{u_\psi(y)-u_\psi(x)}{|u_\psi(y)-u_\psi(x)|}dm_x(y)\);
\item[(iii)] \(u\) is a function of \(m\)-least gradient in \(\Omega\), i.e., \(TV_m(u)\leq TV_m(u+g)\) for every \(g\in BV_m(X)\) with \(g\equiv0\) \(\,\nu\)-a.e. on \(X\setminus\Omega\);
\item[(iv)] \(u\big|_{\Omega_m}\) is a solution of the \(m\)-least gradient problem
\begin{align*}
\min\left\{TV_m(u)\,\,:\,\, u\in BV_m(\Omega_m)\,\,\text{such that}\,\, u=\psi \,\,\nu\text{-a.s. on}\,\partial_m\Omega \right\}.
\end{align*}
\end{itemize}
Further, the authors present several examples of \((m,\nu)\) satisfying a \(p\)-Poincaré inequality as well as some counterexamples.
Reviewer: Yana Kinderknecht (Saarbrücken)Random walk on random planar maps: spectral dimension, resistance and displacementhttps://www.zbmath.org/1483.600692022-05-16T20:40:13.078697Z"Gwynne, Ewain"https://www.zbmath.org/authors/?q=ai:gwynne.ewain"Miller, Jason"https://www.zbmath.org/authors/?q=ai:miller.jason-pSummary: We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d.. increments or a two-dimensional Brownian motion via a ``mating-of-trees'' type bijection. This class includes the uniform infinite planar triangulation (UIPT), the infinite-volume limits of random planar maps weighted by the number of spanning trees, bipolar orientations, or Schnyder woods they admit, and the \(\gamma \)-mated-CRT map for \(\gamma \in (0,2)\). For each of these maps, we obtain an upper bound for the Green's function on the diagonal, an upper bound for the effective resistance to the boundary of a metric ball, an upper bound for the return probability of the random walk to its starting point after \(n\) steps, and a lower bound for the graph-distance displacement of the random walk, all of which are sharp up to polylogarithmic factors.
When combined with work of \textit{J. R. Lee} [Ann. Probab. 49, No. 6, 2671--2731 (2021; Zbl 1482.05319)], our bound for the return probability shows that the spectral dimension of each of these random planar maps is a.s. equal to 2, that is, the (quenched) probability that the simple random walk returns to its starting point after \(2n\) steps is \({n^{-1+{o_n}(1)}}\). Our results also show that the amount of time that it takes a random walk to exit a metric ball is at least its volume (up to a polylogarithmic factor). In the special case of the UIPT, this implies that random walk typically travels at least \({n^{1/4-{o_n}(1)}}\) units of graph distance in \(n\) units of time. The matching upper bound for the displacement is proven by the first author and \textit{T. Hutchcroft} [Probab. Theory Relat. Fields 178, No. 1--2, 567--611 (2020; Zbl 1471.60160)]. These two works together resolve a conjecture of \textit{I. Benjamini} and \textit{N. Curien} [Geom. Funct. Anal. 23, No. 2, 501--531 (2013; Zbl 1274.60143)] in the UIPT case.
Our proofs are based on estimates for the mated-CRT map (which come from its relationship to SLE-decorated Liouville quantum gravity) and a strong coupling of the mated-CRT map with the other random planar map models.Upper tail asymptotics for the intersection local times of random walks in high dimensionshttps://www.zbmath.org/1483.600702022-05-16T20:40:13.078697Z"Mörters, Peter"https://www.zbmath.org/authors/?q=ai:morters.peterSummary: In high dimensions two independent simple random walks have only a finite number of intersections. I describe the main result obtained in a joint paper with \textit{X. Chen} [J. Lond. Math. Soc., II. Ser. 79, No. 1, 186--210 (2009; Zbl 1170.60019)] in which we determine the exact upper tail behaviour of the intersection local time.Convolution algebra for extended Feller convolutionhttps://www.zbmath.org/1483.600712022-05-16T20:40:13.078697Z"Lee, Wha-Suck"https://www.zbmath.org/authors/?q=ai:lee.wha-suck"Le Roux, Christiaan"https://www.zbmath.org/authors/?q=ai:le-roux.christiaanThe paper considers the continuous analogue to the problem of intertwined Markov processes with extended Chapman-Kolmogorov's equation in the form of two distinct continuous state spaces and two homogeneous Markov processes, which models random transitions within a continuum of ``life'' states and from the ``life'' states to a continuum of ``death'' states. The absorbing barrier is modelled as a continuum as ``death'' states for the case of a single ``death'' state.
To handle two-dimensional uni-directional homogeneous stochastic kernels arising in this problem, the authors use recently introduced framework of admissible homomorphisms in the form of a convolution algebra of $\mathbb{C}^2$-valued admissible homomorphisms. For an adequate operator representation of such kernels, the algebra product, which is a non-commutative extension of the Feller convolution, is needed.
Reviewer: Anatoliy Swishchuk (Calgary)Monte Carlo integration of non-differentiable functions on \([0,1]^\iota\), \(\iota =1,\ldots, d\), using a single determinantal point pattern defined on \([0,1]^d\)https://www.zbmath.org/1483.600722022-05-16T20:40:13.078697Z"Coeurjolly, Jean-François"https://www.zbmath.org/authors/?q=ai:coeurjolly.jean-francois"Mazoyer, Adrien"https://www.zbmath.org/authors/?q=ai:mazoyer.adrien"Amblard, Pierre-Olivier"https://www.zbmath.org/authors/?q=ai:amblard.pierre-olivierSummary: This paper concerns the use of a particular class of determinantal point processes (DPP), a class of repulsive spatial point processes, for Monte Carlo integration. Let \(d\ge 1, I\subseteq \overline{d}=\{1,\ldots, d\}\) with \(\iota =|I|\). Using a single set of \(N\) quadrature points \(\{u_1, \ldots, u_N\}\) defined, once for all, in dimension \(d\) from the realization of a specific DPP, we investigate ``minimal'' assumptions on the integrand in order to obtain unbiased Monte Carlo estimates of \(\mu (f_I)=\int_{[0,1]^{\iota}}f_I(u)\text{d}u\) for any known \(\iota\)-dimensional integrable function on \([0,1]^\iota\). In particular, we show that the resulting estimator has variance with order \(N^{-1-(2s\wedge 1)/d}\) when the integrand belongs to some Sobolev space with regularity \(s>0\). When \(s>1/ 2\) (which includes a large class of non-differentiable functions), the variance is asymptotically explicit and the estimator is shown to satisfy a Central Limit Theorem.On the multiplicative chaos of non-Gaussian log-correlated fieldshttps://www.zbmath.org/1483.600732022-05-16T20:40:13.078697Z"Junnila, Janne"https://www.zbmath.org/authors/?q=ai:junnila.janneSummary: We study non-Gaussian log-correlated multiplicative chaos, where the random field is defined as a sum of independent fields that satisfy suitable moment and regularity conditions. The convergence, existence of moments, and analyticity with respect to the inverse temperature are proven for the resulting chaos in the full subcritical range. These results are generalizations of the corresponding theorems for Gaussian multiplicative chaos. A basic example where our results apply is the non-Gaussian Fourier series
\[
\sum\limits_{k=1}^\infty\frac{1}{\sqrt{k}}(A_k \cos(2\pi k x)+B_k \sin(2\pi k x)),
\]
where \(A_k\) and \(B_k\) are i.i.d. random variables.Percolation of the excursion sets of planar symmetric shot noise fieldshttps://www.zbmath.org/1483.600742022-05-16T20:40:13.078697Z"Lachieze-Rey, Raphael"https://www.zbmath.org/authors/?q=ai:lachieze-rey.raphael"Muirhead, Stephen"https://www.zbmath.org/authors/?q=ai:muirhead.stephenSummary: We prove the existence of phase transitions in the global connectivity of the excursion sets of planar symmetric shot noise fields. Our main result establishes a phase transition with respect to the level for shot noise fields with symmetric log-concave mark distributions, including Gaussian, uniform, and Laplace marks, and kernels that are positive, symmetric, and have sufficient tail decay. Without the log-concavity assumption we prove a phase transition with respect to the intensity of positive marks.Correction to: ``A third representation of Feynman-Kac-Itô formula with singular magnetic vector potential''https://www.zbmath.org/1483.600752022-05-16T20:40:13.078697Z"Murayama, Taro"https://www.zbmath.org/authors/?q=ai:murayama.taroCorrection to the author's paper [ibid. 111, No. 2, Paper No. 33, 21 p. (2021; Zbl 1480.60143)].Estimates of the difference between two probability densities of Wiener functionals and its applicationhttps://www.zbmath.org/1483.600762022-05-16T20:40:13.078697Z"Cao, Guilan"https://www.zbmath.org/authors/?q=ai:cao.guilan"He, Kai"https://www.zbmath.org/authors/?q=ai:he.kaiSummary: This study investigates precise estimates of the difference between two probability densities of Wiener functionals in the space of continuously differentiable functions and the Hölder continuous functions. As an application, the convergence rate of the density of the solution to non-Markovian stochastic differential equations is derived utilizing these precise estimates.Switching problems with controlled randomisation and associated obliquely reflected BSDEshttps://www.zbmath.org/1483.600772022-05-16T20:40:13.078697Z"Bénézet, Cyril"https://www.zbmath.org/authors/?q=ai:benezet.cyril"Chassagneux, Jean-François"https://www.zbmath.org/authors/?q=ai:chassagneux.jean-francois"Richou, Adrien"https://www.zbmath.org/authors/?q=ai:richou.adrienSummary: We introduce and study a new class of optimal switching problems, namely \textit{switching problem with controlled randomisation}, where some extra-randomness impacts the choice of switching modes and associated costs. We show that the optimal value of the switching problem is related to a new class of multidimensional obliquely reflected BSDEs. These BSDEs allow as well to construct an optimal strategy and thus to solve completely the initial problem. The other main contribution of our work is to prove new existence and uniqueness results for these obliquely reflected BSDEs. This is achieved by a careful study of the domain of reflection and the construction of an appropriate oblique reflection operator in order to invoke results from \textit{J.-F. Chassagneux} and \textit{A. Richou} [Ann. Inst. Henri Poincaré, Probab. Stat. 56, No. 4, 2868--2896 (2020; Zbl 1478.60172)].Mean-field backward-forward stochastic differential equations and nonzero sum stochastic differential gameshttps://www.zbmath.org/1483.600782022-05-16T20:40:13.078697Z"Chen, Yinggu"https://www.zbmath.org/authors/?q=ai:chen.yinggu"Djehiche, Boualem"https://www.zbmath.org/authors/?q=ai:djehiche.boualem"Hamadène, Said"https://www.zbmath.org/authors/?q=ai:hamadene.saidRandom quasi-periodic paths and quasi-periodic measures of stochastic differential equationshttps://www.zbmath.org/1483.600792022-05-16T20:40:13.078697Z"Feng, Chunrong"https://www.zbmath.org/authors/?q=ai:feng.chunrong"Qu, Baoyou"https://www.zbmath.org/authors/?q=ai:qu.baoyou"Zhao, Huaizhong"https://www.zbmath.org/authors/?q=ai:zhao.huaizhongSummary: In this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic measures for stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariant measure by considering lifted flow and semigroup on cylinder and the tightness of the average of lifted quasi-periodic measures. We further prove that the invariant measure is unique, and thus ergodic.On solutions of stochastic equations with current and osmotic velocitieshttps://www.zbmath.org/1483.600802022-05-16T20:40:13.078697Z"Gliklikh, Yuri E."https://www.zbmath.org/authors/?q=ai:gliklikh.yuri-eSummary: The main aim of this paper is to collect the results connected with existence of solutions of equations with current and osmotic velocities, published in several articles, and supply all constructions and results with complete proofs. Some new properties of both types of equations and their interrelation are described.
For the entire collection see [Zbl 1470.46003].General fully coupled FBSDES involving the value function and related nonlocal HJB equations combined with algebraic equationshttps://www.zbmath.org/1483.600812022-05-16T20:40:13.078697Z"Hao, Tao"https://www.zbmath.org/authors/?q=ai:hao.tao"Zhu, Qingfeng"https://www.zbmath.org/authors/?q=ai:zhu.qingfengStochastic spikes and Poisson approximation of one-dimensional stochastic differential equations with applications to continuously measured quantum systemshttps://www.zbmath.org/1483.600822022-05-16T20:40:13.078697Z"Kolb, Martin"https://www.zbmath.org/authors/?q=ai:kolb.martin"Liesenfeld, Matthias"https://www.zbmath.org/authors/?q=ai:liesenfeld.matthiasSummary: Motivated by the recent contribution [\textit{M. Bauer} and \textit{D. Bernard}, Ann. Henri Poincaré 19, No. 3, 653--693 (2018; Zbl 1391.60073)], we study the scaling limit behavior of a class of one-dimensional stochastic differential equations which has a unique attracting point subject to a small additional repulsive perturbation. Problems of this type appear in the analysis of continuously monitored quantum systems. We extend the results of Bauer and Bernard [loc. cit.] and prove a general result concerning the convergence to a homogeneous Poisson process using only classical probabilistic tools.On time inhomogeneous stochastic Itô equations with drift in \(L_{D+1}\)https://www.zbmath.org/1483.600832022-05-16T20:40:13.078697Z"Krylov, N. V."https://www.zbmath.org/authors/?q=ai:krylov.nicolai-vThe author studies the solvability of Itô stochastic equations with a uniformly nondegenerate bounded measurable diffusion and drift in \(L_{d+1}(\mathbb{R}^d)\). The author proves the existence of weak solutions for the Itô stochastic equations. Furthermore, he proves the existence of strong Markov processes corresponding to the diffusion and drift with the properties indicated above. The main technical tools are collected in Section 4, where the author proves new mixed norm estimates for the distributions of semimartingales.
Reviewer: Feng Chen (Changchun)Harnack and shift Harnack inequalities for degenerate (functional) stochastic partial differential equations with singular driftshttps://www.zbmath.org/1483.600842022-05-16T20:40:13.078697Z"Lv, Wujun"https://www.zbmath.org/authors/?q=ai:lv.wujun"Huang, Xing"https://www.zbmath.org/authors/?q=ai:huang.xingSummary: The existence and uniqueness of the mild solutions for a class of degenerate functional stochastic partial differential equations (SPDEs) are obtained, where the drift is assumed to be Hölder-Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack inequality is derived by the method of coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case. An example is presented in the final part of the paper.Backward stochastic differential equations driven by \(G\)-Brownian motion with uniformly continuous generatorshttps://www.zbmath.org/1483.600852022-05-16T20:40:13.078697Z"Wang, Falei"https://www.zbmath.org/authors/?q=ai:wang.falei"Zheng, Guoqiang"https://www.zbmath.org/authors/?q=ai:zheng.guoqiangSummary: The present paper is devoted to investigating the existence and uniqueness of solutions to a class of non-Lipschitz scalar-valued backward stochastic differential equations driven by \(G\)-Brownian motion. In fact, when the generators are Lipschitz continuous in \(y\) and uniformly continuous in \(z\), we construct the unique solution to such equations by a linearization technique and a monotone convergence argument. The comparison theorem and related nonlinear Feynman-Kac formula are stated as well.The elliptic stochastic quantization of some two dimensional Euclidean QFTshttps://www.zbmath.org/1483.600862022-05-16T20:40:13.078697Z"Albeverio, Sergio"https://www.zbmath.org/authors/?q=ai:albeverio.sergio-a"De Vecchi, Francesco C."https://www.zbmath.org/authors/?q=ai:de-vecchi.francesco-carlo"Gubinelli, Massimiliano"https://www.zbmath.org/authors/?q=ai:gubinelli.massimilianoSummary: We study a class of elliptic SPDEs with additive Gaussian noise on \(\mathbb{R}^2\times M\), with \(M\) a \(d\)-dimensional manifold equipped with a positive Radon measure, and a real-valued non linearity given by the derivative of a smooth potential \(V\), convex at infinity and growing at most exponentially. For quite general coefficients and a suitable regularity of the noise we obtain, via the dimensional reduction principle discussed in our previous paper [Ann. Probab. 48, No. 4, 1693--1741 (2020; Zbl 1446.60041)], the identity between the law of the solution to the SPDE evaluated at the origin with a Gibbs type measure on an abstract Wiener space over \(M\). The results are then applied to the elliptic stochastic quantization equation for the scalar field with polynomial interaction over \(\mathbb{T}^2\), and with exponential interaction over \(\mathbb{R}^2\) (known also as Høegh-Krohn or Liouville model in the literature). In particular for the exponential interaction case, the existence and uniqueness properties of solutions to the elliptic equation over \(\mathbb{R}^{2+2}\) is derived as well as the dimensional reduction for the values of the ``charge parameter'' \(\sigma=\frac{\alpha}{2\sqrt{\pi}}< \sqrt{4(8-4\sqrt{3})\pi}\simeq\sqrt{4.29\pi}\), for which the model has an Euclidean invariant, reflection positive probability measure (hence also permitting to get the corresponding relativistic invariant model on the two dimensional Minkowski space).Hermite spatial variations for the solution to the stochastic heat equationhttps://www.zbmath.org/1483.600872022-05-16T20:40:13.078697Z"Araya, Héctor"https://www.zbmath.org/authors/?q=ai:araya.hector"Garzón, Johanna"https://www.zbmath.org/authors/?q=ai:garzon.johanna"Moreno, Nicolás"https://www.zbmath.org/authors/?q=ai:moreno.nicolas"Plaza, Francisco"https://www.zbmath.org/authors/?q=ai:plaza.franciscoThis paper studies the asymptotic behavior of Hermite spatial variations for the solution to the stochastic heat equation (SHE) with additive space-time white noise, where the spatial dimensional is one. Under suitable normalization, the central limit theorem (CLT) and almost sure central limit theorem (ASCLT) for the sequence of Hermite variations are obtained by the use of Malliavin calculus, Stein's method and the explicit expression for the solution to SHE.
Reviewer: Lifeng Chen (Shanghai)The impact of white noise on a supercritical bifurcation in the Swift-Hohenberg equationhttps://www.zbmath.org/1483.600882022-05-16T20:40:13.078697Z"Bianchi, Luigi Amedeo"https://www.zbmath.org/authors/?q=ai:bianchi.luigi-amedeo"Blömker, Dirk"https://www.zbmath.org/authors/?q=ai:blomker.dirkSummary: We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation. Using an approximation via modulation equations, we provide a tool to analyse how the noise influences the dynamics close to a change of stability.Strong solution to stochastic penalised nematic liquid crystals model driven by multiplicative Gaussian noisehttps://www.zbmath.org/1483.600892022-05-16T20:40:13.078697Z"Brzeźniak, Zdzisław"https://www.zbmath.org/authors/?q=ai:brzezniak.zdzislaw"Hausenblas, Erika"https://www.zbmath.org/authors/?q=ai:hausenblas.erika"Razafimandimby, Paul André"https://www.zbmath.org/authors/?q=ai:razafimandimby.paul-andreSummary: In this paper, we prove the existence of unique maximal local strong solutions to a stochastic system for both 2D and 3D penalised nematic liquid crystals driven by multiplicative Gaussian noise. In the 2D case, we show this solution is global. As a by-product of our investigation, but of independent interest, we present a general method based on fixed-point arguments to establish the existence and uniqueness of a maximal local solution of abstract stochastic evolution equations with coefficients satisfying a local Lipschitz condition involving the norms of two different Banach spaces.Incompressible viscous fluids in \(\mathbb{R}^2\) and SPDEs on graphs, in presence of fast advection and non smooth noisehttps://www.zbmath.org/1483.600902022-05-16T20:40:13.078697Z"Cerrai, Sandra"https://www.zbmath.org/authors/?q=ai:cerrai.sandra"Xi, Guangyu"https://www.zbmath.org/authors/?q=ai:xi.guangyuSummary: The asymptotic behavior of a class of stochastic reaction-diffusion-advection equations in the plane is studied. We show that as the divergence-free advection term becomes larger and larger, the solutions of such equations converge to the solution of a suitable stochastic PDE defined on the graph associated with the Hamiltonian. Firstly, we deal with the case that the stochastic perturbation is given by a singular spatially homogeneous Wiener process taking values in the space of Schwartz distributions. As in previous works, we assume here that the derivative of the period of the motion on the level sets of the Hamiltonian does not vanish. Then, in the second part, without assuming this condition on the derivative of the period, we study a weaker type of convergence for the solutions of a suitable class of linear SPDEs.Spatial ergodicity for SPDEs via Poincaré-type inequalitieshttps://www.zbmath.org/1483.600912022-05-16T20:40:13.078697Z"Chen, Le"https://www.zbmath.org/authors/?q=ai:chen.le"Khoshnevisan, Davar"https://www.zbmath.org/authors/?q=ai:khoshnevisan.davar"Nualart, David"https://www.zbmath.org/authors/?q=ai:nualart.david"Pu, Fei"https://www.zbmath.org/authors/?q=ai:pu.feiSummary: Consider a parabolic stochastic PDE of the form \(\partial_t u=\frac{1}{2}\Delta u+\sigma (u)\eta\), where \(u=u(t,x)\) for \(t\geq 0\) and \(x\in\mathbb{R}^d, \sigma \colon\mathbb{R}\to \mathbb{R}\) is Lipschitz continuous and non random, and \(\eta\) is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation \(f\). If, in addition, \(u(0)\equiv 1\), then we prove that, under a mild decay condition on \(f\), the process \(x\mapsto u(t,x)\) is stationary and ergodic at all times \(t>0\). It has been argued that, when coupled with moment estimates, spatial ergodicity of \(u\) teaches us about the intermittent nature of the solution to such SPDEs [\textit{L. Bertini} and \textit{N. Cancrini}, J. Stat. Phys. 78, No. 5--6, 1377--1401 (1995; Zbl 1080.60508); \textit{D. Khoshnevisan}, Analysis of stochastic partial differential equations. Providence, RI: American Mathematical Society (AMS) (2014; Zbl 1304.60005)]. Our results provide rigorous justification of such discussions.
Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincaré inequalities. We further showcase the utility of these Poincaré inequalities by: (a) describing conditions that ensure that the random field \(u(t)\) is mixing for every \(t> 0\); and by (b) giving a quick proof of a conjecture of \textit{D. Conus} et al. [Electron. J. Probab. 17, Paper No. 102, 15 p. (2012; Zbl 1296.60165)] about the ``size'' of the intermittency islands of \(u\).
The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of \textit{G. Maruyama} [Mem. Fac. Sci. Kyūsyū Univ., Ser. A 4, 45--106 (1949; Zbl 0045.40602)] (see also [\textit{H. Dym} and \textit{H. P. McKean}, Gaussian processes, function theory, and the inverse spectral problem. Probability and Mathematical Statistics. Vol. 31. New York San Francisco - London: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers. (1976; Zbl 0327.60029)]) in the simple setting where the nonlinear term \(\sigma\) is a constant function.Study of nonlinear stochastic Cauchy problems in \((\mathcal{C},\mathcal{E},\mathcal{P})\)-algebrashttps://www.zbmath.org/1483.600922022-05-16T20:40:13.078697Z"Dévoué, Victor"https://www.zbmath.org/authors/?q=ai:devoue.victorSummary: We use the framework of the \((\mathcal{C},\mathcal{E},\mathcal{P})\)-algebras of \textit{J.-A. Marti} [Chapman Hall/CRC Res. Notes Math. 401, 175--186 (1999; Zbl 0938.35008)] to study some nonlinear stochastic Cauchy problems for a simple equation, namely the transport equation in basic form, with stochastic generalized processes. Until now such studies were made in Colombeau-type algebras.Moments of the 2D SHE at criticalityhttps://www.zbmath.org/1483.600932022-05-16T20:40:13.078697Z"Gu, Yu"https://www.zbmath.org/authors/?q=ai:gu.yu.1"Quastel, Jeremy"https://www.zbmath.org/authors/?q=ai:quastel.jeremy"Tsai, Li-Cheng"https://www.zbmath.org/authors/?q=ai:tsai.li-chengSummary: We study the stochastic heat equation in two spatial dimensions with a multiplicative white noise, as the limit of the equation driven by a noise that is mollified in space and white in time. As the mollification radius \(\varepsilon\to 0 \), we tune the coupling constant near the critical point, and show that the single time correlation functions converge to a limit written in terms of an explicit nontrivial semigroup. Our approach consists of two steps. First we show the convergence of the resolvent of the (tuned) two-dimensional delta Bose gas, by adapting the framework of \textit{J. Dimock} and \textit{S. G. Rajeev} [J. Phys. A, Math. Gen. 37, No. 39, 9157--9173 (2004; Zbl 1067.81024)] to our setup of spatial mollification. Then we match this to the Laplace transform of our semigroup.Stochastic heat equation with general rough noisehttps://www.zbmath.org/1483.600942022-05-16T20:40:13.078697Z"Hu, Yaozhong"https://www.zbmath.org/authors/?q=ai:hu.yaozhong"Wang, Xiong"https://www.zbmath.org/authors/?q=ai:wang.xiongSummary: We study the well-posedness of a nonlinear one dimensional stochastic heat equation driven by Gaussian noise: \(\frac{\partial u}{\partial t}=\frac{{\partial^2}u}{\partial{x^2}}+\sigma (u)\dot{W}\), where \(\dot{W}\) is white in time and fractional in space with Hurst parameter \(H\in(\frac{1}{4},\frac{1}{2})\). In a recent paper [Ann. Probab. 45, No. 6B, 4561--4616 (2017; Zbl 1393.60066)] by \textit{Y. Hu} et al. a technical and unusual condition of \(\sigma (0)=0\) was assumed which is critical in their approach. The main effort of this paper is to remove this condition. The idea is to work on a weighted space \(\mathcal{Z}_{\lambda,T}^p\) for some power decay weight \(\lambda(x)={c_H}(1+|x|^2)^{H-1}\). In addition, when \(\sigma(u)=1\) we obtain the exact asympotics of the solution \(u_{\mathrm{add}}(t,x)\) as \(t\) and \(x\) go to infinity. In particular, we find the exact growth of \(\sup_{|x|\le L}|{u_{\text{add}}}(t,x)|\) and the sharp growth rate for the Hölder coefficients, namely, \({\sup_{|x|\le L}}\frac{|{u_{\text{add}}}(t,x+h)-{u_{\text{add}}}(t,x)|}{|h{|^{\beta }}}\) and \({\sup_{|x|\le L}}\frac{|{u_{\text{add}}}(t+\tau ,x)-{u_{\text{add}}}(t,x)|}{{\tau^{\alpha}}}\).Propagation of singularities for the stochastic wave equationhttps://www.zbmath.org/1483.600952022-05-16T20:40:13.078697Z"Lee, Cheuk Yin"https://www.zbmath.org/authors/?q=ai:lee.cheuk-yin"Xiao, Yimin"https://www.zbmath.org/authors/?q=ai:xiao.yiminThe authors study the existence and propagation of singularities of the solution to a one-dimensional linear stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. In this article, singularities are associated with the law of the iterated logarithm (LIL), or the local modulus of continuity. The authors refer to the random points at which the process exhibits local oscillations that are much larger than those given by the LIL. For Brownian motion, this phenomenon was first studied by \textit{S. Orey} and \textit{S. J. Taylor} [Proc. Lond. Math. Soc. (3) 28, 174--192 (1974; Zbl 0292.60128)].
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Global well-posedness of 2D stochastic Burgers equations with multiplicative noisehttps://www.zbmath.org/1483.600962022-05-16T20:40:13.078697Z"Zhou, Guoli"https://www.zbmath.org/authors/?q=ai:zhou.guoli"Wang, Lidan"https://www.zbmath.org/authors/?q=ai:wang.lidan"Wu, Jiang-Lun"https://www.zbmath.org/authors/?q=ai:wu.jianglunThis paper deals with the existence and uniqueness of a solution to the 2D stochastic Burger equation given by
\[
\begin{aligned}
du(t)&=\nu\Delta u dt-(u\cdot\nabla)udt+u(t)\circ dW(t), \\
u(t,x)&=0, \quad t>0, \quad x=(x_1,x_2)\in\partial D.\\
u(0,x)&=u_0(x),\quad x=(x_1,x_2).
\end{aligned} \tag{1}
\]
Here, \(D\subset\mathbb{R}^2\) is a smooth, bounded, open domain, \(W=\sum_{k=1}^\infty\sigma_kB_k(t)\) with \(\sum_{k=1}^\infty\sigma_k^2<\infty\) where \((B_k)_{k\in\mathbb{N}}\) is a sequence of independent, Wiener processes. The parameter \(\nu>0\) stands for the viscosity and \(\circ\) denotes the Stratonovich integral.
The authors prove that for any initial condition \(u_0\in\mathbb{H}^1\), such equation \((1)\) has a unique global strong solution \(v\) and \(v\in C([0,T];\mathbb{H}^1)\cap L^2([0,T];\mathbb{H}^1)\). \(\mathbb{H}^1\) denotes the domain of the operator \((-\Delta)^{1/2}\).
Reviewer: Adrián Hinojosa-Calleja (Barcelona)A probabilistic interpretation of conservation and balance lawshttps://www.zbmath.org/1483.600972022-05-16T20:40:13.078697Z"Belopolskaya, Ya. I."https://www.zbmath.org/authors/?q=ai:belopolskaya.yana-iSummary: We construct stochastic systems associated with parabolic conservation and balance laws written in the form of nonlinear parabolic systems and prove that solution of stochastic systems allows to construct weak and mild solutions of the Cauchy problem for original partial differential equations (PDE) partial differential equations systems.
For the entire collection see [Zbl 1470.46003].Strong stochastic persistence of some Lévy-driven Lotka-Volterra systemshttps://www.zbmath.org/1483.600982022-05-16T20:40:13.078697Z"Videla, Leonardo"https://www.zbmath.org/authors/?q=ai:videla.leonardo-aSummary: We study a class of Lotka-Volterra stochastic differential equations with continuous and pure-jump noise components, and derive conditions that guarantee the strong stochastic persistence (SSP) of the populations engaged in the ecological dynamics. More specifically, we prove that, under certain technical assumptions on the jump sizes and rates, there is convergence of the laws of the stochastic process to a unique stationary distribution supported far away from extinction. We show how the techniques and conditions used in proving SSP for general Kolmogorov systems driven solely by Brownian motion must be adapted and tailored in order to account for the jumps of the driving noise. We provide examples of applications to the case where the underlying food-web is: (a) a \(1\)-predator, \(2\)-prey food-web, and (b) a multi-layer food-chain.Lanchester model with the random coefficientshttps://www.zbmath.org/1483.600992022-05-16T20:40:13.078697Z"Zadorozhniy, V. G."https://www.zbmath.org/authors/?q=ai:zadorozhnii.v-gSummary: A mathematical model of Lanchester's combat operations is studied in the form of a system of differential equations, whose coefficients are random processes. It is assumed that the random coefficients are given by the characteristic functional. The first and second moment functions of system solutions are found. The problem is reduced to a deterministic system of differential equations with ordinary and variational derivatives. Gaussian and uniformly distributed random coefficients are considered.
For the entire collection see [Zbl 1470.46003].Multiple Markov Gaussian processeshttps://www.zbmath.org/1483.601002022-05-16T20:40:13.078697Z"Kowalski, Zbigniew S."https://www.zbmath.org/authors/?q=ai:kowalski.zbigniew-s.1Summary: We get a necessary and sufficient condition on the density of the spectral measure for stationary Gaussian processes with a discrete set of parameters to be Markov of order \(k\). We introduce a natural definition of the Markov property of order \(r\in \mathbb{R}_+,\) in the case of continuous parameter. Moreover we give an extension of multiple Markov Gaussian processes with discrete parameters to Gaussian semiflows with some weaker property than the multiple Gaussian property.Rapid mixing of the switch Markov chain for 2-class joint degree matriceshttps://www.zbmath.org/1483.601012022-05-16T20:40:13.078697Z"Amanatidis, Georgios"https://www.zbmath.org/authors/?q=ai:amanatidis.georgios"Kleer, Pieter"https://www.zbmath.org/authors/?q=ai:kleer.pieterMixing and average mixing times for general Markov processeshttps://www.zbmath.org/1483.601022022-05-16T20:40:13.078697Z"Anderson, Robert M."https://www.zbmath.org/authors/?q=ai:anderson.robert-m"Duanmu, Haosui"https://www.zbmath.org/authors/?q=ai:duanmu.haosui"Smith, Aaron"https://www.zbmath.org/authors/?q=ai:smith.aaron-m|smith.aaron-carlSummary: \textit{Y. Peres} and \textit{P. Sousi} [J. Theor. Probab. 28, No. 2, 488--519 (2015; Zbl 1323.60094)] showed that the mixing times and average mixing times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on compact state spaces that satisfy the strong Feller property.Metastability in a lattice gas with strong anisotropic interactions under Kawasaki dynamicshttps://www.zbmath.org/1483.601032022-05-16T20:40:13.078697Z"Baldassarri, Simone"https://www.zbmath.org/authors/?q=ai:baldassarri.simone"Nardi, Francesca Romana"https://www.zbmath.org/authors/?q=ai:nardi.francesca-romanaSummary: In this paper we analyze metastability and nucleation in the context of a local version of the Kawasaki dynamics for the two-dimensional strongly anisotropic Ising lattice gas at very low temperature. Let \(\Lambda = \{ 0,1,\ldots,L\}^2 \subset \mathbb{Z}^2\) be a finite box. Particles perform simple exclusion on \(\Lambda\), but when they occupy neighboring sites they feel a binding energy \(-U_1 < 0\) in the horizontal direction and \(-U_2 < 0\) in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume \(\Lambda\). Along each bond touching the boundary of \(\Lambda\) from the outside to the inside, particles are created with rate \(\rho =e^{-\Delta\beta}\), while along each bond from the inside to the outside, particles are annihilated with rate 1, where \(\beta\) is the inverse temperature and \(\Delta > 0\) is an activity parameter. Thus, the boundary of \(\Lambda\) plays the role of an infinite gas reservoir with density \(\rho\). We consider the parameter regime \(U_1 > 2U_2\) also known as the strongly anisotropic regime. We take \(\Delta \in (U_1,U_1 +U_2)\) and we prove that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We consider the asymptotic regime corresponding to finite volume in the limit of large inverse temperature \(\beta\). We investigate how the transition from empty to full takes place. In particular, we estimate in probability, expectation and distribution the asymptotic transition time from the metastable configuration to the stable configuration. Moreover, we identify the size of the \textit{critical droplets}, as well as some of their properties. For the weakly anisotropic model corresponding to the parameter regime \(U_1 < 2U_2\), analogous results have already been obtained. We observe very different behavior in the weakly and strongly anisotropic regimes. We find that the \textit{Wulff shape}, i.e., the shape minimizing the energy of a droplet at fixed volume, is not relevant for the critical configurations.The Wright-Fisher model for class-dependent fitness landscapeshttps://www.zbmath.org/1483.601042022-05-16T20:40:13.078697Z"Dalmau, Joseba"https://www.zbmath.org/authors/?q=ai:dalmau.josebaSummary: We consider a population evolving under mutation and selection. The genotype of an individual is a word of length \(\ell\) over a finite alphabet. Mutations occur during reproduction, independently on each locus; the fitness depends on the Hamming class (the distance to a reference sequence \(w^{\ast})\). Evolution is driven according to the classical Wright-Fisher process. We focus on the proportion of the different classes under the invariant measure of the process. We consider the regime where the length of the genotypes \(\ell\) goes to infinity, and
\[
\text{population size }\sim \ell,\qquad\text{mutation rate }\sim 1/ \ell.
\]
We prove the existence of a critical curve, which depends both on the population size and the mutation rate. Below the critical curve, the proportion of any fixed class converges to 0, whereas above the curve, it converges to a positive quantity, for which we give an explicit formula.Ergodicity coefficients for higher-order stochastic processeshttps://www.zbmath.org/1483.601052022-05-16T20:40:13.078697Z"Fasino, Dario"https://www.zbmath.org/authors/?q=ai:fasino.dario"Tudisco, Francesco"https://www.zbmath.org/authors/?q=ai:tudisco.francescoReconstructing a recurrent random environment from a single trajectory of a random walk in random environment with errorshttps://www.zbmath.org/1483.601062022-05-16T20:40:13.078697Z"Jalowy, Jonas"https://www.zbmath.org/authors/?q=ai:jalowy.jonas"Löwe, Matthias"https://www.zbmath.org/authors/?q=ai:lowe.matthiasSummary: We consider one infinite path of a Random Walk in Random Environment (RWRE, for short) in an unknown environment. This environment consists of either i.i.d. site or bond randomness. At each position the random walker stops and tells us the environment it sees at the point where it is, without telling us, where it is. These observations \({\chi^{\prime }}\) are spoiled by reading errors that occur with probability \(p<1\). We show: If the RWRE is recurrent and satisfies the standard assumptions on such RWREs, then with probability one in the environment, the errors, and the random walk we are able reconstruct the law of the environment. For most situations this result is even independent of the value of \(p\). If the distribution of the environment has a non-atomic part, we can even reconstruct the environment itself, up to translation.On the continuous dual Hahn processhttps://www.zbmath.org/1483.601072022-05-16T20:40:13.078697Z"Bryc, Włodek"https://www.zbmath.org/authors/?q=ai:bryc.wlodzimierzThe author extends the continuous dual Hahn process \((\mathbb{T}_t)\) of \textit{I. Corwin} and \textit{A. Knizel} [``Stationary measure for the open KPZ equation'', Preprint, \url{arXiv:2103.12253}] from a finite time interval to the entire real line by taking a limit of a closely related Markov process \((T_t).\) The processes \((T_t) \) are characterized by conditional means and variances under bidirectional conditioning, and it is proved that continuous dual Hahn polynomials are orthogonal martingale polynomials for both processes.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Matrix analysis for continuous-time Markov chainshttps://www.zbmath.org/1483.601082022-05-16T20:40:13.078697Z"Le, Hung V."https://www.zbmath.org/authors/?q=ai:le.hung-viet"Tsatsomeros, M. J."https://www.zbmath.org/authors/?q=ai:tsatsomeros.michael-jSummary: Continuous-time Markov chains have transition matrices that vary continuously in time. Classical theory of nonnegative matrices, M-matrices and matrix exponentials is used in the literature to study their dynamics, probability distributions and other stochastic properties. For the benefit of Perron-Frobenius cognoscentes, this theory is surveyed and further adapted to study continuous-time Markov chains on finite state spaces.Co-jumps and Markov counting systems in random environmentshttps://www.zbmath.org/1483.601092022-05-16T20:40:13.078697Z"Bretó, Carles"https://www.zbmath.org/authors/?q=ai:breto.carlesSummary: Motivated by the analysis of multi-strain infectious disease data, we provide closed-form transition rates for continuous-time Markov chains that arise from subjecting Markov counting systems to correlated environmental noises. Noise correlation induces co-jumps or counts that occur simultaneously in several counting processes. Such co-jumps are necessary and sufficient for infinitesimal correlation between counting processes of the system. We analyzed such infinitesimal correlation for a specific infectious disease model by randomizing time of Kolmogorov's Backward system of differential equations based on appropriate stochastic integrals.
For the entire collection see [Zbl 1470.53006].A sharp log-Sobolev inequality for the multislicehttps://www.zbmath.org/1483.601102022-05-16T20:40:13.078697Z"Salez, Justin"https://www.zbmath.org/authors/?q=ai:salez.justinThis paper studies a sharp log-Sobolev inequality for the multislice. Consider a list of positive integers \(\kappa=(\kappa_1,\kappa_2,\cdots,\kappa_L)\) for some \(L\ge2\) and \(n=\kappa_1+\kappa_2+\cdots+\kappa_L\). Let \(\tau_{LS}(\kappa)\) be the optimal values in the corresponding functional inequality known as the log-Sobolev constant. Let \(\kappa_{\min}\) be the minimum number of the list \(\kappa\). The main result of the paper is a log-Sobolev constant of the multislice. For all values of \(\kappa\), the following inequalities hold: \(\log(n/\kappa_{\min})\le\tau_{LS}(\kappa)\le(4/\log2)\log(n/\kappa_{\min})\). The sharpness of constant is proved. The pre-factor in front of the logarithm is sharp.
Reviewer: Yilun Shang (Newcastle)Concentration of scalar ergodic diffusions and some statistical implicationshttps://www.zbmath.org/1483.601112022-05-16T20:40:13.078697Z"Aeckerle-Willems, Cathrine"https://www.zbmath.org/authors/?q=ai:aeckerle-willems.cathrine"Strauch, Claudia"https://www.zbmath.org/authors/?q=ai:strauch.claudiaSummary: We derive uniform concentration inequalities for continuous-time analogues of empirical processes and related stochastic integrals of scalar ergodic diffusion processes. Thereby, we lay the foundation typically required for the study of sup-norm properties of estimation procedures for a large class of diffusion processes. In the classical i.i.d. context, a key device for the statistical sup-norm analysis is provided by Talagrand-type concentration inequalities. Aiming for a parallel substitute in the diffusion framework, we present a systematic, self-contained approach to such uniform concentration inequalities via martingale approximation and moment bounds obtained by the generic chaining method. The developed machinery is of independent probabilistic interest and can serve as a starting point for investigations of other processes such as more general Markov processes, in particular multivariate or discretely observed diffusions. As a first concrete statistical application, we analyse the sup-norm error of estimating the invariant density of an ergodic diffusion via the local time estimator and the classical nonparametric kernel density estimator, respectively.Covariant Symanzik identitieshttps://www.zbmath.org/1483.601122022-05-16T20:40:13.078697Z"Kassel, Adrien"https://www.zbmath.org/authors/?q=ai:kassel.adrien"Lévy, Thierry"https://www.zbmath.org/authors/?q=ai:levy.thierrySummary: Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of a large class of random, in general non-Gaussian, fields in terms of holonomies of random paths with respect to an annealed random gauge field, in the spirit of Symanzik's foundational work on the subject.One-dimensional Wiener process with the properties of partial reflection and delayhttps://www.zbmath.org/1483.601132022-05-16T20:40:13.078697Z"Kopytko, B. I."https://www.zbmath.org/authors/?q=ai:kopytko.bogdan-i"Shevchuk, R. V."https://www.zbmath.org/authors/?q=ai:shevchuk.r-vSummary: In this paper, we construct the two-parameter semigroup of operators associated with a certain one-dimensional inhomogeneous diffusion process and study its properties. We are interested in the process on the real line which can be described as follows. At the interior points of the half-lines separated by a point, the position of which depends on the time variable, this process coincides with the Wiener process given there and its behavior on the common boundary of these half-lines is determined by a kind of the conjugation condition of Feller-Wentzell's type. The conjugation condition we consider is local and contains only the first-order derivatives of the unknown function with respect to each of its variables.
The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding conjugation problem for a second order linear parabolic equation to which the above problem is reduced. Its classical solvability is obtained by the boundary integral equations method under the assumption that the initial function is bounded and continuous on the whole real line, the parameters characterizing the Feller-Wentzell conjugation condition are continuous functions of the time variable, and the curve defining the common boundary of the domains is determined by the function which is continuously differentiable and its derivative satisfies the Hölder condition with exponent less than \(1/2\).Duality for maximum entropy diffusion MRIhttps://www.zbmath.org/1483.601142022-05-16T20:40:13.078697Z"Maréchal, Pierre"https://www.zbmath.org/authors/?q=ai:marechal.pierreSummary: We derive an entropy model for diffusion MRI and show that Fenchel duality techniques make this model tractable. This generalizes a model proposed by Daniel Alexander in 2005, in which the displacement of particles is confined to a sphere. In order to better suit the physics of the diffusion process, we propose to relax this constraint. The Kullback-Leibler relative entropy is used to measure the discrepancy between the probability to be inferred and some reference measure. The obtained optimization problem is then studied using tools from partially finite convex progamming. The solution can be computed via the unconstrained maximization of a smooth concave function, whose number of variable is merely (twice) the number of Fourier samples.
For the entire collection see [Zbl 1470.00021].On occupation times of one-dimensional diffusionshttps://www.zbmath.org/1483.601152022-05-16T20:40:13.078697Z"Salminen, Paavo"https://www.zbmath.org/authors/?q=ai:salminen.paavo-h"Stenlund, David"https://www.zbmath.org/authors/?q=ai:stenlund.davidSummary: In this paper, we study the moment generating function and the moments of occupation time functionals of one-dimensional diffusions. Assuming, specifically, that the process lives on \(\mathbb{R}\) and starts at \(0\), we apply Kac's moment formula and the strong Markov property to derive an expression for the moment generating function in terms of the Green kernel of the underlying diffusion. Moreover, the approach allows us to derive a recursive equation for the Laplace transforms of the moments of the occupation time on \(\mathbb{R}_+\). If the diffusion has a scaling property, the recursive equation simplifies to an equation for the moments of the occupation time up to time \(1\). As examples of diffusions with scaling property, we study in detail skew two-sided Bessel processes and, as a special case, skew Brownian motion. It is seen that for these processes our approach leads to simple explicit formulas. The recursive equation for a sticky Brownian motion is also discussed.On recurrent properties of Fisher-Wright's diffusion on \((0,1)\) with mutationhttps://www.zbmath.org/1483.601162022-05-16T20:40:13.078697Z"Sineokiy, Roman"https://www.zbmath.org/authors/?q=ai:sineokiy.roman"Veretennikov, Alexander"https://www.zbmath.org/authors/?q=ai:veretennikov.alexander-yuThis short paper proves a refined exponential recurrent bound for a one-dimensional Fisher-Wright diffusion process living on the interval \((0,1)\) subject to mutations. This bound implies an exponential rate of convergence towards the invariant measure.
More precisely, consider the following one-dimensional stochastic differential equation
\[
dX_t =\left[\#1\right]{a(1-X_t) -bX_t}dt + \varepsilon \sqrt{X_t\left(\#1\right){1-X_t}}dW_t
\]
with parameters \(a,b,\varepsilon >0\). The starting point \(X_0=x\in (0,1)\) is deterministic and \((W_t)\) is a one dimensional standard Brownian motion (under \({\mathbb P}_x\)).
Such equations were introduced for the study of population genetics independently by Wright and Fisher and remain a topical area of investigations until now. In the context of the paper, the model is subject to mutations and the parameters \(a\) and \(b\) stand for the mutation rates (for an account on such models see [\textit{L. Chen} and \textit{D. W. Stroock}, SIAM J. Math. Anal. 42, No. 2, 539--567 (2010; Zbl 1221.35013); \textit{C. L. Epstein} and \textit{R. Mazzeo}, SIAM J. Math. Anal. 42, No. 2, 568--608 (2010; Zbl 1221.35063)] in the authors' reference list) and \(\varepsilon\) as a selection parameter.
Well-known classical results on stochastic differential equations ensure that there is a pathwise unique strong solution for this equation and that this solution is a strong Markov process. Since \((0,1)\) is topologically equivalent to \(\mathbb R\), one may apply Feller's test to show that this solution remains in the open interval \((0,1)\) for all times whenever the condition \(\min(a,b)>\varepsilon^2/2\) is satisfied (Feller's condition).
Although one may wish to apply general results ensuring the existence of an invariant measure with an exponential rate of convergence for general one-dimensional models (see for, e.g., [\textit{L. H. Duc} et al., Stochastic Processes Appl. 128, No. 10, 3253--3272 (2018; Zbl 1434.60138)]), the strategy is quite different here and the main result of the paper gives a deeper insight on the recurrent properties of the Fisher-Wright diffusion process with a more detailed explicit result for the recurrence properties of the solution.
More precisely, for any \(\alpha\in (0,1)\) let \(\tau_\alpha = \inf\left(\#1\right){t\geq 0~:~X_t\in [\alpha, 1-\alpha]}\) stand for the time where the solution enters \([\alpha, 1-\alpha]\). Then, assuming that Feller's condition is satisfied, the main result of the paper asserts that for any constant \(c>0\), there exists a point \(\alpha\in (0,1/2)\) and \(m>0\) such that
\[
\mathbb E_x \mathrm{e}^{c\tau_\alpha} \leq C(m)\,c\,\alpha^{m+1}\left(\#1\right){(1-x)^{-m} + x^{-m}} + 1
\]
with \(C(m) = \frac{2}{\min(a,b)m - \varepsilon^2m(m+1)/2}\).
In a nutshell, the methodology of proof is to draw clever consequences of the application of Itô's formula to the function \(v(t,x) = x^{-m}{\exp}(ct)\).
The stated inequality ensures that there are exponential moments for the returns from very small neighborhood of \(0\) or \(1\) to a compact set in \((0,1)\) and moreover that these exponential moments are quantified by an explicit bound derived from the data.
As a by product of their main result, the authors recover the exponential rate of convergence of the distribution of \(X_t\) towards the invariant measure w.r.t the total variation metric (the computations are not written in the text but are announced for a forthcoming research study concerning more general diffusions).
Note also that the paper contains an independent and short proof that the solution remains \({\mathbb P}_x\)-a.s. in \((0,1)\) whenever Feller's condition is satisfied which is inspired by \textit{I. I. Gikhhman} [``A short remark on Feller's square root condition'', Preprint, \url{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1756450}].
\textbf{NB}: all number references are given by the authors' reference list.
Reviewer: Miguel Martinez (Marne-la-Vallée)A result on the Laplace transform associated with the sticky Brownian motion on an intervalhttps://www.zbmath.org/1483.601172022-05-16T20:40:13.078697Z"Song, Shiyu"https://www.zbmath.org/authors/?q=ai:song.shiyuTotal number of births on the negative half-line of the binary branching Brownian motion in the boundary casehttps://www.zbmath.org/1483.601182022-05-16T20:40:13.078697Z"Chen, Xinxin"https://www.zbmath.org/authors/?q=ai:chen.xinxin"Mallein, Bastien"https://www.zbmath.org/authors/?q=ai:mallein.bastienSummary: The binary branching Brownian motion in the boundary case is a particle system on the real line behaving as follows. It starts with a unique particle positioned at the origin at time \(0\). The particle moves according to a Brownian motion with drift \(\mu =2\) and diffusion coefficient \(\sigma^2=2\), until an independent exponential time of parameter \(1\). At that time, the particle dies giving birth to two children who then start independent copies of the same process from their birth place. It is well-known that in this system, the cloud of particles eventually drifts to \(\infty\). The aim of this note is to provide a precise estimate for the total number of particles that were born on the negative half-line, investigating in particular the tail decay of this random variable.Local time and local reflection of the Wiener processhttps://www.zbmath.org/1483.601192022-05-16T20:40:13.078697Z"Ibragimov, I. A."https://www.zbmath.org/authors/?q=ai:ibragimov.ildar-a"Smorodina, N. V."https://www.zbmath.org/authors/?q=ai:smorodina.natalya-v"Faddeev, M. M."https://www.zbmath.org/authors/?q=ai:faddeev.m-mSummary: In this paper we introduce a concept of a Brownian trajectory local reflection. Ideologically, this concept is close to the concept of the Brownian local time, which can be considered as the integrated (over time) Dirac delta function of a Brownian trajectory. In the concept of the local reflection, we replace the Dirac delta function by its first derivative.
For the entire collection see [Zbl 1470.46003].The inverse first-passage-place problem for Wiener processeshttps://www.zbmath.org/1483.601202022-05-16T20:40:13.078697Z"Lefebvre, Mario"https://www.zbmath.org/authors/?q=ai:lefebvre.marioLet \(X(t)\) be a one-dimensional diffusion with \(X(0) = x \in (0,1)\) and let \(\tau(x)\) be the first exit time from the interval \((0,1)\). The first-passage-place probabilities \(P[X(\tau(x)) = 0] \) are known for a large number of diffusion processes, [\textit{A. N. Borodin} and \textit{P. Salminen}, Handbook of Brownian motion: Facts and formulae. Basel: Birkhäuser (2002; Zbl 1012.60003)]. The inverse first-passage-place problem considered is the following: Let \(q \in (0,1)\) be given. Let \(X(t)\) be a diffusion with \(X(0)\) a random variable on \((0,1)\). If \(\tau\) denotes the first exit time from \((0,1)\), the main task is try to determine the probability distribution of \(X(0)\) such that \(P[X(\tau) = 0] = q \). The solution to this problem is not necessarily unique. The author gives a characterization of the distribution of \(X(0)\) in the case of \(X\) being standard Brownian motion and Brownian motion with drift. A number of examples of distributions solving the inverse problem are given, both discrete and continuous.
Reviewer: Göran Högnäs (Åbo)Uniqueness in law for stable-like processes of variable orderhttps://www.zbmath.org/1483.601212022-05-16T20:40:13.078697Z"Jin, Peng"https://www.zbmath.org/authors/?q=ai:jin.pengSummary: Let \(d\ge 1\). Consider a stable-like operator of variable order
\[
\mathcal{A}f(x)=\int_{{\mathbb{R}}^d\backslash\{0\}}\left[f(x+h)-f(x)-\mathbf{1}_{\{|h|\le 1\}}h\cdot\nabla f(x)\right]n(x,h)|h|^{-d-\alpha(x)}\mathrm{d}h,
\]
where \(0< \inf_x\alpha(x)\le\sup_x\alpha(x)< 2\) and \(n(x, h)\) satisfies
\[
n(x,h)=n(x,-h),\quad 0< \kappa_1\le n(x,h)\le\kappa_2,\quad\forall x,h\in\mathbb{R}^d,
\]
with \(\kappa_1\) and \(\kappa_2\) being some positive constants. Under some further mild conditions on the functions \(n(x, h)\) and \(\alpha (x)\), we show the uniqueness of solutions to the martingale problem for \(\mathcal{A}\).A conditional functional limit theorem for a decomposable branching processhttps://www.zbmath.org/1483.601222022-05-16T20:40:13.078697Z"Afanasyev, V. I."https://www.zbmath.org/authors/?q=ai:afanasev.valerii-ivanovichSummary: A decomposable Galton-Watson branching process with two particle types is studied. It is assumed that a particle of the first type produces equal numbers of particles of the first and second types, while a particle of the second type produces only particles of their own type. Under the condition that the total number of particles of the second type is greater than \(N \rightarrow \infty \), a functional limit theorem for the process describing the number of particles of the second type in different generations is proved.
For the entire collection see [Zbl 1470.46003].A note on asymptotic behavior of critical Galton-Watson processes with immigrationhttps://www.zbmath.org/1483.601232022-05-16T20:40:13.078697Z"Barczy, Mátyás"https://www.zbmath.org/authors/?q=ai:barczy.matyas"Bezdány, Dániel"https://www.zbmath.org/authors/?q=ai:bezdany.daniel"Pap, Gyula"https://www.zbmath.org/authors/?q=ai:pap.gyulaSummary: In this somewhat didactic note we give a detailed alternative proof of the known result of \textit{C. Z. Wei} and \textit{J. Winnicki} [Stochastic Processes Appl. 31, No. 2, 261--282 (1989; Zbl 0673.60092)] which states that, under second-order moment assumptions on the offspring and immigration distributions, the sequence of appropriately scaled random step functions formed from a critical Galton-Watson process with immigration (not necessarily starting from zero) converges weakly towards a squared Bessel process. The proof of Wei and Winnicki [loc. cit.] is based on infinitesimal generators, while we use limit theorems for random step processes towards a diffusion process due to \textit{M. Ispány} and \textit{G. Pap} [Acta Math. Hung. 126, No. 4, 381--395 (2010; Zbl 1274.60109)]. This technique was already used by \textit{M. Ispány} [Publ. Math. 72, No. 1--2, 17--34 (2008; Zbl 1164.60061)], who proved functional limit theorems for a sequence of some appropriately normalized nearly critical Galton-Watson processes with immigration starting from zero, where the offspring means tend to its critical value 1. As a special case of Theorem 2.1 of Ispány [2008, loc. cit.] one can get back the result of Wei and Winnicki [loc. cit.] in the case of zero initial value. In the present note we handle nonzero initial values with the technique used by Ispány [2008, loc. cit.], and further, we simplify some of the arguments in the proof of Theorem 2.1 of Ispány [2008, loc. cit.] as well.Trait-dependent branching particle systems with competition and multiple offspringhttps://www.zbmath.org/1483.601242022-05-16T20:40:13.078697Z"Berzunza, Gabriel"https://www.zbmath.org/authors/?q=ai:berzunza.gabriel"Sturm, Anja"https://www.zbmath.org/authors/?q=ai:sturm.anja-k"Winter, Anita"https://www.zbmath.org/authors/?q=ai:winter.anitaSummary: In this work we model the dynamics of a population that evolves as a continuous time branching process with a trait structure and ecological interactions in form of mutations and competition between individuals. We generalize existing microscopic models by allowing individuals to have multiple offspring at a reproduction event. Furthermore, we allow the reproduction law to be influenced both by the trait type of the parent as well as by the mutant trait type.
We look for tractable large population approximations. More precisely, under some natural assumption on the branching and mutation mechanisms, we establish a superprocess limit as the solution of a well-posed martingale problem. Standard approaches do not apply in our case due to the lack of the branching property, which is a consequence of the dependency created by the competition between individuals. In order to show uniqueness we therefore had to develop a generalization of Dawson's Girsanov Theorem that may be of independent interest.Scaling limits of tree-valued branching random walkshttps://www.zbmath.org/1483.601252022-05-16T20:40:13.078697Z"Duquesne, Thomas"https://www.zbmath.org/authors/?q=ai:duquesne.thomas"Khanfir, Robin"https://www.zbmath.org/authors/?q=ai:khanfir.robin"Lin, Shen"https://www.zbmath.org/authors/?q=ai:lin.shen.1|lin.shen"Torri, Niccolò"https://www.zbmath.org/authors/?q=ai:torri.niccoloSummary: We consider a branching random walk (BRW) taking its values in the \(\mathtt{b}\)-ary rooted tree \(\mathbb{W}_{\mathtt{b}}\) (i.e. the set of finite words written in the alphabet \(\{ 1,\ldots,\mathtt{b}\}\), with \(\mathtt{b}\geq 2)\). The BRW is indexed by a critical Galton-Watson tree conditioned to have \(n\) vertices; its offspring distribution is aperiodic and is in the domain of attraction of a \(\gamma\)-stable law, \(\gamma \in (1,2]\). The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on \(\mathbb{W}_{\mathtt{b}}\) (reflection at the root of \(\mathbb{W}_{\mathtt{b}}\) and otherwise: probability \(1/2\) to move closer to the root of \(\mathbb{W}_{\mathtt{b}}\) and probability \(1/(2\mathtt{b})\) to move away from it to one of the \(\mathtt{b}\) sites above). We denote by \(\mathcal{R}_{\mathtt{b}} (n)\) the range of the BRW in \(\mathbb{W}_{\mathtt{b}}\) which is the set of all sites in \(\mathbb{W}_{\mathtt{b}}\) visited by the BRW. We first prove a law of large numbers for \(\# \mathcal{R}_{\mathtt{b}} (n)\) and we also prove that if we equip \(\mathcal{R}_{\mathtt{b}}(n)\) (which is a random subtree of \(\mathbb{W}_{\mathtt{b}})\) with its graph-distance \(d_{\mathtt{gr}}\), then there exists a scaling sequence \((a_n)_{n\in \mathbb{N}}\) satisfying \(a_n \to \infty\) such that the metric space \((\mathcal{R}_{\mathtt{b}} (n), a_n^{-1} d_{\mathtt{gr}})\), equipped with its normalised empirical measure, converges to the reflected Brownian cactus with \(\gamma\)-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by \textit{N. Curien} et al. [Ann. Inst. Henri Poincaré, Probab. Stat. 49, No. 2, 340--373 (2013; Zbl 1275.60035)].Rank dependent branching-selection particle systemshttps://www.zbmath.org/1483.601262022-05-16T20:40:13.078697Z"Groisman, Pablo"https://www.zbmath.org/authors/?q=ai:groisman.pablo"Soprano-Loto, Nahuel"https://www.zbmath.org/authors/?q=ai:soprano-loto.nahuelSummary: We consider a large family of branching-selection particle systems. The branching rate of each particle depends on its rank and is given by a function \(b\) defined on the unit interval. There is also a killing measure \(D\) supported on the unit interval as well. At branching times, a particle is chosen among all particles to the left of the branching one by sampling its rank according to \(D\). The measure \(D\) is allowed to have total mass less than one, which corresponds to a positive probability of no killing. Between branching times, particles perform independent Brownian Motions in the real line. This setting includes several well known models like Branching Brownian Motion (BBM), \(N\)-BBM, rank dependent BBM, and many others. We conjecture a scaling limit for this class of processes and prove such a limit for a related class of branching-selection particle systems. This family is rich enough to allow us to use the behavior of solutions of the limiting equation to prove the asymptotic velocity of the rightmost particle under minimal conditions on \(b\) and \(D\). The behavior turns out to be universal and depends only on \(b(1)\) and the total mass of \(D\). If the total mass is one, the number of particles in the system \(N\) is conserved and the velocities \(v_N\) converge to \(\sqrt{2b(1)}\). When the total mass of \(D\) is less than one, the number of particles in the system grows up in time exponentially fast and the asymptotic velocity of the rightmost one is \(\sqrt{2b(1)}\) independently of the number of initial particles.Gaussian fluctuations and a law of the iterated logarithm for Nerman's martingale in the supercritical general branching processhttps://www.zbmath.org/1483.601272022-05-16T20:40:13.078697Z"Iksanov, Alexander"https://www.zbmath.org/authors/?q=ai:iksanov.aleksander-m"Kolesko, Konrad"https://www.zbmath.org/authors/?q=ai:kolesko.konrad"Meiners, Matthias"https://www.zbmath.org/authors/?q=ai:meiners.matthiasSummary: In his, by now, classical work from [Z. Wahrscheinlichkeitstheor. Verw. Geb. 57, 365--395 (1981; Zbl 0451.60078)], \textit{O. Nerman} made extensive use of a crucial martingale \((W_t)_{t\geq 0}\) to prove convergence in probability, in mean and almost surely, of supercritical general branching processes (also known as Crump-Mode-Jagers branching processes) counted with a general characteristic. The martingale terminal value \(W\) figures in the limits of his results.
We investigate the rate at which the martingale, now called \textit{Nerman's martingale}, converges to its limit \(W\). More precisely, assuming the existence of a Malthusian parameter \(\alpha > 0\) and \(W_0\in L^2\), we prove a functional central limit theorem for \((W-W_{t+s})_{s\in \mathbb{R}}\), properly normalized, as \(t\to \infty\). The weak limit is a randomly scaled time-changed Brownian motion. Under an additional technical assumption, we prove a law of the iterated logarithm for \(W-W_t\).Subcritical branching processes in random environment with immigration stopped at zerohttps://www.zbmath.org/1483.601282022-05-16T20:40:13.078697Z"Li, Doudou"https://www.zbmath.org/authors/?q=ai:li.doudou"Vatutin, Vladimir"https://www.zbmath.org/authors/?q=ai:vatutin.vladimir-a"Zhang, Mei"https://www.zbmath.org/authors/?q=ai:zhang.meiSummary: We consider the subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are change of measure and some conditional limit theorems for random walks.Influence of the configuration of particle generation sources on the behavior of branching walks: a case studyhttps://www.zbmath.org/1483.601292022-05-16T20:40:13.078697Z"Yarovaya, E. B."https://www.zbmath.org/authors/?q=ai:yarovaya.elena-bSummary: We consider a supercritical continuous-time branching random walk on a multidimensional lattice with finite number of particle generation sources of the same intensities without any restrictions on the variance of jumps of the underlying random walk. The effect of ``limit coalescence'' of eigenvalues is revealed for an arrangement of sources under which the pairwise distances between them go off to infinity. The effect of the arrangement of particle generation sources on the order of appearance of positive eigenvalues in the spectrum of the evolutionary operator with receding sources is revealed.
For the entire collection see [Zbl 1470.46003].On a multivariate generalized Polya process without regularity propertyhttps://www.zbmath.org/1483.601302022-05-16T20:40:13.078697Z"Cha, Ji Hwan"https://www.zbmath.org/authors/?q=ai:cha.ji-hwan"Badía, F. G."https://www.zbmath.org/authors/?q=ai:badia.francisco-germanSummary: Most of the multivariate counting processes studied in the literature are regular processes, which implies, ignoring the types of the events, the non-occurrence of multiple events. However, in practice, several different types of events may occur simultaneously. In this paper, a new class of multivariate counting processes which allow simultaneous occurrences of multiple types of events is suggested and its stochastic properties are studied. For the modeling of such kind of process, we rely on the tool of superposition of seed counting processes. It will be shown that the stochastic properties of the proposed class of multivariate counting processes are explicitly expressed. Furthermore, the marginal processes are also explicitly obtained. We analyze the multivariate dependence structure of the proposed class of counting processes.Dispatching to parallel servers. Solutions of Poisson's equation for first-policy improvementhttps://www.zbmath.org/1483.601312022-05-16T20:40:13.078697Z"Bilenne, Olivier"https://www.zbmath.org/authors/?q=ai:bilenne.olivierSummary: Policy iteration techniques for multiple-server dispatching rely on the computation of value functions. In this context, we consider the continuous-space \(M/G/1\)-FCFS queue endowed with an arbitrarily designed cost function for the waiting times of the incoming jobs. The associated relative value function is a solution of Poisson's equation for Markov chains, which in this work we solve in the Laplace transform domain by considering an ancillary, underlying stochastic process extended to (imaginary) negative backlog states. This construction enables us to issue closed-form relative value functions for polynomial and exponential cost functions and for piecewise compositions of the latter, in turn permitting the derivation of interval bounds for the relative value function in the form of power series or trigonometric sums. We review various cost approximation schemes and assess the convergence of the interval bounds these induce on the relative value function, namely Taylor expansions (divergent, except for a narrow class of entire functions with low orders of growth) and uniform approximation schemes (polynomials, trigonometric), which achieve optimal convergence rates over finite intervals. This study addresses all the steps to implementing dispatching policies for systems of parallel servers, from the specification of \textit{general} cost functions toward the computation of interval bounds for the relative value functions and the exact implementation of the first-policy improvement step.Exponential ergodicity and steady-state approximations for a class of Markov processes under fast regime switchinghttps://www.zbmath.org/1483.601322022-05-16T20:40:13.078697Z"Arapostathis, Ari"https://www.zbmath.org/authors/?q=ai:arapostathis.aristotle"Pang, Guodong"https://www.zbmath.org/authors/?q=ai:pang.guodong"Zheng, Yi"https://www.zbmath.org/authors/?q=ai:zheng.yiSummary: We study ergodic properties of a class of Markov-modulated general birth-death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the `averaged' fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth-death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.Shot-noise queueing modelshttps://www.zbmath.org/1483.601332022-05-16T20:40:13.078697Z"Boxma, Onno"https://www.zbmath.org/authors/?q=ai:boxma.onno-j"Mandjes, Michel"https://www.zbmath.org/authors/?q=ai:mandjes.michelSummary: We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results are derived for the workload in an \(M/G/1\) shot-noise queue and some of its variants. Furthermore, we give some attention to queues with general workload-dependent service speed. We also discuss linear stochastic fluid networks, and queues in which the input process is a shot-noise process.Performance measures for the two-node queue with finite buffershttps://www.zbmath.org/1483.601342022-05-16T20:40:13.078697Z"Chen, Yanting"https://www.zbmath.org/authors/?q=ai:chen.yanting"Bai, Xinwei"https://www.zbmath.org/authors/?q=ai:bai.xinwei"Boucherie, Richard J."https://www.zbmath.org/authors/?q=ai:boucherie.richard-j"Goseling, Jasper"https://www.zbmath.org/authors/?q=ai:goseling.jasperSummary: We consider a two-node queue modeled as a two-dimensional random walk. In particular, we consider the case that one or both queues have finite buffers. We develop an approximation scheme based on the Markov reward approach to error bounds in order to bound performance measures of such random walks. The approximation scheme is developed in terms of a perturbed random walk in which the transitions along the boundaries are different from those in the original model and the invariant measure of the perturbed random walk is of product-form. We then apply this approximation scheme to a tandem queue and some variants of this model, for the case that both buffers are finite. The modified approximation scheme and the corresponding applications for a two-node queueing system in which only one of the buffers has finite capacity have also been discussed.A random access G-network: stability, stable throughput, and queueing analysishttps://www.zbmath.org/1483.601352022-05-16T20:40:13.078697Z"Dimitriou, Ioannis"https://www.zbmath.org/authors/?q=ai:dimitriou.ioannis"Pappas, Nikolaos"https://www.zbmath.org/authors/?q=ai:pappas.nikolaos-dSummary: The effect of signals on stability, stable throughput region, and delay in a two-user slotted ALOHA-based random-access system with collisions is considered. This work gives rise to the development of random access G-networks, which can model security attacks, expiration of deadlines, or other malfunctions, and introduce load balancing among highly interacting queues. The users are equipped with infinite capacity buffers accepting external bursty arrivals. We consider both negative and triggering signals. Negative signals delete a packet from a user queue, while triggering signals cause the instantaneous transfer of packets among user queues. We obtain the exact stability region, and show that the stable throughput region is a subset of it. Moreover, we perform a compact mathematical analysis to obtain exact expressions for the queueing delay by solving a non-homogeneous Riemann boundary value problem. A computationally efficient way to obtain explicit bounds for the expected number of buffered packets at user queues is also presented. The theoretical findings are numerically evaluated and insights regarding the system performance are derived.Equilibrium balking strategies in the repairable \(M/M/1\) \(G\)-retrial queue with complete removalshttps://www.zbmath.org/1483.601362022-05-16T20:40:13.078697Z"Gao, Shan"https://www.zbmath.org/authors/?q=ai:gao.shan"Zhang, Deran"https://www.zbmath.org/authors/?q=ai:zhang.deran"Dong, Hua"https://www.zbmath.org/authors/?q=ai:dong.hua"Wang, Xianchao"https://www.zbmath.org/authors/?q=ai:wang.xianchaoSummary: We consider an \(M/M/1\) retrial queue subject to negative customers (called as \(G\)-retrial queue). The arrival of a negative customer forces all positive customers to leave the system and causes the server to fail. At a failure instant, the server is sent to be repaired immediately. Based on a natural reward-cost structure, all arriving positive customers decide whether to join the orbit or balk when they find the server is busy. All positive customers are selfish and want to maximize their own net benefit. Therefore, this system can be modeled as a symmetric noncooperative game among positive customers and the fundamental problem is to identify the Nash equilibrium balking strategy, which is a stable strategy in the sense that if all positive customers agree to follow it no one can benefit by deviating from it, that is, it is a strategy that is the best response against itself. In this paper, by using queueing theory and game theory, the Nash equilibrium mixed strategy in unobservable case and the Nash equilibrium pure strategy in observable case are considered. We also present some numerical examples to demonstrate the effect of the information together with some parameters on the equilibrium behaviors.Finding nonstationary state probabilities of open Markov networks with multiple classes of customers and various featureshttps://www.zbmath.org/1483.601372022-05-16T20:40:13.078697Z"Matalytski, Mikhail"https://www.zbmath.org/authors/?q=ai:matalytski.mikhail"Kopats, Dmitry"https://www.zbmath.org/authors/?q=ai:kopats.dmitrySummary: This paper discusses a system of difference-differential equations (DDE) that is satisfied by the time-dependent state probabilities of open Markov queueing networks with various features. The number of network states in this case and the number of equations in this system is infinite. Flows of customers arriving at the network are a simple and independent, the time of customer services is exponentially distributed. The intensities of transitions between the network states are deterministic functions depending on its states.
To solve the system of DDE, we propose a modified method of successive approximations, combined with the method of series. The convergence of successive approximations with time to a stationary probability distribution, the form of which is indicated in the paper has been proved. The sequence of approximations converges to a unique solution of the system of equations. Any successive approximation can be represented as a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for calculations on a computer. Examples of the analysis of Markov G-networks with various features have been presented.On the extreme values of \(M/M/m\) queueing systemshttps://www.zbmath.org/1483.601382022-05-16T20:40:13.078697Z"Matsak, Ivan K."https://www.zbmath.org/authors/?q=ai:matsak.ivan-kThe paper under review considers a stationary and ergodic \(M/M/m\) queueing system. Let \(t_0=0, t_1, \dots\) be the arrival moments, and let \(W_n\) denote the waiting time of the \(n\)th arriving customer at the moment of arrival, \(W_0=0\). The author establishes asymptotic theorems for the probability distributions of the following characteristics: \(\overline{W}(t)=\sup_{0\leq t_k\leq t}W_k(s)\), \(\overline{W}_n=\overline{W}(t_n)=\max_{1\leq k\leq n}W_k\) as \(t\to\infty\) and \(n\to\infty\).
Reviewer: Vyacheslav Abramov (Melbourne)Product-form Markovian queueing systems with multiple resourceshttps://www.zbmath.org/1483.601392022-05-16T20:40:13.078697Z"Naumov, Valeriy"https://www.zbmath.org/authors/?q=ai:naumov.valerii-a"Samouylov, Konstantin"https://www.zbmath.org/authors/?q=ai:samouylov.konstantin-eSummary: In the paper, we study general Markovian models of loss systems with random resource requirements, in which customers at arrival occupy random quantities of various resources and release them at departure. Customers may request negative quantities of resources, but total amount of resources allocated to customers should be nonnegative and cannot exceed predefined maximum levels. Allocating a negative volume of a resource to a customer leads to a temporary increase in its volume in the system. We derive necessary and sufficient conditions for the product-form of the stationary probability distribution of the Markov jump process describing the system.Kelly and Jackson networks with interchangeable, cooperative servershttps://www.zbmath.org/1483.601402022-05-16T20:40:13.078697Z"Wang, Chia-Li"https://www.zbmath.org/authors/?q=ai:wang.chiali"Wolff, Ronald W."https://www.zbmath.org/authors/?q=ai:wolff.ronald-wSummary: In open Kelly and Jackson networks, servers are assigned to individual stations, serving customers only where they are assigned. We investigate the performance of modified networks where servers cooperate. A server who would be idle at the assigned station will serve customers at another station, speeding up service there. We assume \textit{interchangeable} servers: the service rate of a server at a station depends only on the station, not the server. This gives \textit{work conservation}, which is used in various ways. We investigate three levels of server cooperation, from \textit{full cooperation}, where all servers are busy when there is work to do anywhere in the network, to \textit{one-way cooperation}, where a server assigned to one station may assist a server at another, but not the converse. We obtain the same stability conditions for each level and, in a series of examples, obtain substantial performance improvement with server cooperation, even when stations before modification are moderately loaded.Equilibrium analysis of observable express service with customer choicehttps://www.zbmath.org/1483.601412022-05-16T20:40:13.078697Z"Zhou, Jiaqi"https://www.zbmath.org/authors/?q=ai:zhou.jiaqi"Ryzhov, Ilya O."https://www.zbmath.org/authors/?q=ai:ryzhov.ilya-oSummary: We study a stylized queueing model motivated by paid express lanes on highways. There are two parallel, observable first-come, first-served queues with finitely many servers: one queue has a faster service rate, but charges a fee to join, and the other is free but slow. Upon arrival, customers see the state of each queue and choose between them by comparing the respective disutility of time spent waiting, subject to random shocks. This framework encompasses both the multinomial logit and exponomial customer choice models. Using a fluid limit approximation, we give a detailed characterization of the equilibrium in this system. We show that social welfare is optimized when the express queue is exactly at (but not over) full capacity; however, in some cases, revenue is maximized by artificially creating congestion in the free queue. The latter behavior is caused by changes in the price elasticity of demand as the service capacity of the free queue fills up.Complete resource pooling of a load-balancing policy for a network of battery swapping stationshttps://www.zbmath.org/1483.601422022-05-16T20:40:13.078697Z"Sloothaak, Fiona"https://www.zbmath.org/authors/?q=ai:sloothaak.fiona"Cruise, James"https://www.zbmath.org/authors/?q=ai:cruise.james-r|cruise.james-f"Shneer, Seva"https://www.zbmath.org/authors/?q=ai:shneer.seva"Vlasiou, Maria"https://www.zbmath.org/authors/?q=ai:vlasiou.maria"Zwart, Bert"https://www.zbmath.org/authors/?q=ai:zwart.bert-pSummary: To reduce carbon emission in the transportation sector, there is currently a steady move taking place to an electrified transportation system. This brings about various issues for which a promising solution involves the construction and operation of a battery swapping infrastructure rather than in-vehicle charging of batteries. In this paper, we study a closed Markovian queueing network that allows for spare batteries under a dynamic arrival policy. We propose a provisioning rule for the capacity levels and show that these lead to near-optimal resource utilization, while guaranteeing good quality-of-service levels for electric vehicle users. Key in the derivations is to prove a state-space collapse result, which in turn implies that performance levels are as good as if there would have been a single station with an aggregated number of resources, thus achieving complete resource pooling.Diffusive bounds for the critical density of activated random walkshttps://www.zbmath.org/1483.601432022-05-16T20:40:13.078697Z"Asselah, Amine"https://www.zbmath.org/authors/?q=ai:asselah.amine"Rolla, Leonardo T."https://www.zbmath.org/authors/?q=ai:rolla.leonardo-t"Schapira, Bruno"https://www.zbmath.org/authors/?q=ai:schapira.brunoSummary: We consider symmetric activated random walks on \(\mathbb{Z}\), and show that the critical density \(\zeta_c\) satisfies \(c \sqrt{\lambda}\leqslant \zeta_c (\lambda) \leqslant C \sqrt{\lambda}\) for small \(\lambda\), where \(\lambda\) denotes the sleep rate.Local stationarity in exponential last-passage percolationhttps://www.zbmath.org/1483.601442022-05-16T20:40:13.078697Z"Balázs, Márton"https://www.zbmath.org/authors/?q=ai:balazs.marton.1"Busani, Ofer"https://www.zbmath.org/authors/?q=ai:busani.ofer"Seppäläinen, Timo"https://www.zbmath.org/authors/?q=ai:seppalainen.timoSummary: We consider point-to-point last-passage times to every vertex in a neighbourhood of size \(\delta N^{2/3}\) at distance \(N\) from the starting point. The increments of the last-passage times in this neighbourhood are shown to be jointly equal to their stationary versions with high probability that depends only on \(\delta \). Through this result we show that (1) the \(\text{Airy}_2\) process is locally close to a Brownian motion in total variation; (2) the tree of point-to-point geodesics from every vertex in a box of side length \(\delta N^{2/3}\) going to a point at distance \(N\) agrees inside the box with the tree of semi-infinite geodesics going in the same direction; (3) two point-to-point geodesics started at distance \(N^{2/3}\) from each other, to a point at distance \(N\), will not coalesce close to either endpoint on the scale \(N\). Our main results rely on probabilistic methods only.The fractal cylinder process: existence and connectivity phase transitionshttps://www.zbmath.org/1483.601452022-05-16T20:40:13.078697Z"Broman, Erik I."https://www.zbmath.org/authors/?q=ai:broman.erik-ivar"Elias, Olof"https://www.zbmath.org/authors/?q=ai:elias.olof"Mussini, Filipe"https://www.zbmath.org/authors/?q=ai:mussini.filipe"Tykesson, Johan"https://www.zbmath.org/authors/?q=ai:tykesson.johan-haraldSummary: We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension \(d\ge 2\), and a connectivity phase transition whenever \(d\ge 4\). We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point.
A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this restriction results in a fractal ellipsoid model which we describe in detail, as it is key to obtaining our main results.
In addition we also determine the almost sure Hausdorff dimension of the fractal set.Slow convergence of Ising and spin glass models with well-separated frustrated verticeshttps://www.zbmath.org/1483.601462022-05-16T20:40:13.078697Z"Gillman, David"https://www.zbmath.org/authors/?q=ai:gillman.david-w|gillman.david-saul|gillman.david-w.1"Randall, Dana"https://www.zbmath.org/authors/?q=ai:randall.dana-jSummary: Many physical models undergo phase transitions as some parameter of the system is varied. This phenomenon has bearing on the convergence times for local Markov chains walking among the configurations of the physical system. One of the most basic examples of this phenomenon is the ferromagnetic Ising model on an \(n\times n\) square lattice region Lambda with mixed boundary conditions. For this spin system, if we fix the spins on the top and bottom sides of the square to be \(+\) and the left and right sides to be \(-\), a standard Peierls argument based on energy shows that below some critical temperature \(t_c\), any local Markov chain \(\mathcal{M}\) requires time exponential in \(n\) to mix.\par Spin glasses are magnetic alloys that generalize the Ising model by specifying the strength of nearest neighbor interactions on the lattice, including whether they are ferromagnetic or antiferromagnetic. Whenever a face of the lattice is bounded by an odd number of edges with ferromagnetic interactions, the face is considered frustrated because the local competing objectives cannot be simultaneously satisfied. We consider spin glasses with exactly four well-separated frustrated faces that are symmetric around the center of the lattice region under 90 degree rotations. We show that local Markov chains require exponential time for all spin glasses in this class. This class includes the ferromagnetic Ising model with mixed boundary conditions described above, where the frustrated faces are on the boundary. The standard Peierls argument breaks down when the frustrated faces are on the interior of \(\Lambda\) and yields weaker results when they are on the boundary of \(\Lambda\) but not near the corners. We show that there is a universal temperature \(T\) below which \(\mathcal{M}\) will be slow for all spin glasses with four well-separated frustrated faces. Our argument shows that there is an exponentially small cut indicated by the free energy, carefully exploiting both entropy and energy to establish a small bottleneck in the state space to establish slow mixing.
For the entire collection see [Zbl 1390.68020].A note on once reinforced random walk on ladder \(\mathbb{Z} \times \{0, 1 \} \)https://www.zbmath.org/1483.601472022-05-16T20:40:13.078697Z"Huang, Xiangyu"https://www.zbmath.org/authors/?q=ai:huang.xiangyu"Liu, Yong"https://www.zbmath.org/authors/?q=ai:liu.yong.4|liu.yong.3|liu.yong.5|liu.yong.1|liu.yong.2"Sidoravicius, Vladas"https://www.zbmath.org/authors/?q=ai:sidoravicius.vladas"Xiang, Kainan"https://www.zbmath.org/authors/?q=ai:xiang.kai-nanSummary: Given any \(\delta \in(0, \infty)\), let \((X_n)_{n = 0}^{\infty}\) be the \(\delta \)-once reinforced random walk on ladder \(\mathbb{Z} \times \{0, 1 \}\) with the following edge weight function at the \((n + 1)\)-th step:
\[
w_n(e) = 1 +(\delta - 1) \cdot I_{\{ N (e, n) > 0 \}} =
\begin{cases}
1 \quad & \text{if } N (e, n) = 0, \\
\delta & \text{if } N (e, n) > 0.
\end{cases}
\]
Here \(N(e, n) : = \# \{i < n : X_i X_{i + 1} = e \}\) is the number of times that edge \(e\) has been traversed by the walk before time \(n\). It was proved that \(( X_n )_{n = 0}^{\infty}\) is almost surely recurrent for \(\delta > 1 \slash 2\) [\textit{M. Vervoort}, ``Reinforced random walks'', Preprint, \url{https://staff.fnwi.uva.nl/m.r.vervoort/walk.pdf}; \textit{T. Sellke}, Electron. J. Probab. 11, Paper No. 11, 301--310 (2006; Zbl 1113.60048)], while the a.s. recurrence for negative reinforcement factor \(\delta \in(0, 1 \slash 2]\) remained open. In this note, we give an affirmative answer to this question.Scaling limit of dynamical percolation on critical Erdős-Rényi random graphshttps://www.zbmath.org/1483.601482022-05-16T20:40:13.078697Z"Rossignol, Raphaël"https://www.zbmath.org/authors/?q=ai:rossignol.raphaelConsider Erdős-Rényi random graphs \(\mathcal G(n,p)\) obtained from the complete graph with \(n \) vertices by deleting edges independently with probability \(1-p\). As \(n\to \infty\), there is a critical phase when \(p:=p(\lambda, n) =\frac 1 n + \frac \lambda {n^{4/3}}\) for some \(\lambda \in{\mathbb R}\). It was proved in [\textit{L. Addario-Berry} et al., Probab. Theory Relat. Fields 152, No. 3--4, 367--406 (2012; Zbl 1239.05165)] that the critical Erdős-Rényi random graphs with appropriate scalings convergence for the Gromov-Hausdorff topology to \(\mathcal G_\lambda\), a collection of real random graphs. In [\textit{L. Addario-Berry} et al., Ann. Probab. 45, No. 5, 3075--3144 (2017; Zbl 1407.60013)], the convergence was proved for the Gromov-Hausdorff-Prokhorov topology.
In the present paper, a dynamical percolation on critical Erdős-Rényi random graphs \(\mathcal G(n,p)\) is considered. It can be constructed from two Poisson processes associated to each edges of the complete graph. One process determines the times when an edge is added (or kept if there is already one) and the second process determines the times when an edge is deleted (if there is one). A dynamical percolation process can also be constructed on the continuous limit \(\mathcal G_\lambda\). It is shown that under appropriate scalings, this process is the limit of the dynamical percolation process starting from a critical Erdős-Rényi random graph. The appropriate topology is a new intricate version of the Gromov-Hausdorff-Prokhorov distance.
The proofs require a detailed analysis of the structure of a multigraph and rely on the properties of the multiplicative coalescent obtained in [\textit{D. Aldous}, Ann. Probab. 25, No. 2, 812--854 (1997; Zbl 0877.60010)].
Convergence is also established for the coalescence process and for the fragmentation process under appropriate scalings. In the first case, edges are never deleted and a Poisson process determines the times where edges are added whilst in the second case, edges are never added and a Poisson process determines the times where edges are deleted.
Reviewer: Daniel Boivin (Brest)Excess deviations for points disconnected by random interlacementshttps://www.zbmath.org/1483.601492022-05-16T20:40:13.078697Z"Sznitman, Alain-Sol"https://www.zbmath.org/authors/?q=ai:sznitman.alain-solSummary: We consider random interlacements on \(\mathbb{Z}^d\), \(d \ge 3\), when their vacant set is in a strongly percolative regime. Given a large box centered at the origin, we establish an asymptotic upper bound on the exponential rate of decay of the probability that the box contains an excessive fraction \(\nu\) of points that are disconnected by random interlacements from the boundary of a concentric box of double size. As an application, we show that when \(\nu\) is not too large this asymptotic upper bound matches the asymptotic lower bound derived in our work [``On bulk deviations for the local behavior of random interlacements'', Preprint, \url{arXiv:1906.05809}], and the exponential rate of decay is governed by the variational problem in the continuum involving the percolation function of the vacant set of random interlacements that we studied in [Prog. Probab. 77, 775--796 (2021; Zbl 1469.60347)]. This is a further confirmation of the pertinence of this variational problem.Distribution of the random walk conditioned on survival among quenched Bernoulli obstacleshttps://www.zbmath.org/1483.601502022-05-16T20:40:13.078697Z"Ding, Jian"https://www.zbmath.org/authors/?q=ai:ding.jian"Fukushima, Ryoki"https://www.zbmath.org/authors/?q=ai:fukushima.ryoki"Sun, Rongfeng"https://www.zbmath.org/authors/?q=ai:sun.rongfeng"Xu, Changji"https://www.zbmath.org/authors/?q=ai:xu.changjiThe paper under review deals with the localization properties of a simple random walk on \(\mathbb{Z}^d\) (\(d\geq 2\)) in the presence of independent Bernoulli obstacles. These obstacles are (i) hard, in the sense that the walk is killed upon visiting any of the obstacle, and (ii) quenched, with respect to the probability measure that is conditioned on the origin being in an infinite open cluster. It was previously established by two of the authors [\textit{J. Ding} and \textit{C. Xu}, Commun. Math. Phys. 375, No. 2, 949--1001 (2020; Zbl 1440.60083)] that the walk conditioned to survive for a large time \(n\) gets eventually localized in a ball whose radius is deterministic, explicit and of logarithmic order in \(n\), and whose center is random and dependent on the obstacle set. The present paper contains two remarkably sharp results. First, the authors prove that with probability converging to one, as \(n\) goes to infinity, the localization ball is (almost) free from obstacles. Then, they provide a local limit theorem for the position of the random walk conditioned to survive, both at the endpoint of the trajectory and inside the bulk. The corresponding limiting laws are the \(\ell^1\) and \(\ell^2\)-normalized eigenfunctions associated to the principal Dirichlet eigenvalue inside the localization ball. The proof builds upon previously established lemmas (such as the existence of a localization pocket with large eigenvalue inside which a low obstacle density region looks almost like a ball) and new intermediate results, including a ball-clearing lemma. The latter is obtained by estimating the impact of removing obstacles on the principal eigenvalue of the killed random walk inside a ball.
Reviewer: Julien Poisat (Paris)Fokker-Planck equation for Feynman-Kac transform of anomalous processeshttps://www.zbmath.org/1483.601512022-05-16T20:40:13.078697Z"Zhang, Shuaiqi"https://www.zbmath.org/authors/?q=ai:zhang.shuaiqi"Chen, Zhen-Qing"https://www.zbmath.org/authors/?q=ai:chen.zhen-qingSummary: In this paper, we develop a novel and rigorous approach to the Fokker-Planck equation, or Kolmogorov forward equation, for the Feynman-Kac transform of non-Markov anomalous processes. The equation describes the evolution of the density of the anomalous process \(Y_t = X_{E_t}\) under the influence of potentials, where \(X\) is a strong Markov process on a Lusin space \(\mathcal{X}\) that is in weak duality with another strong Markov process \(\widehat{X}\) on \(\mathcal{X}\) and \(\{ E_t, t \geq 0\}\) is the inverse of a driftless subordinator \(S\) that is independent of \(X\) and has infinite Lévy measure. We derive a probabilistic representation of the density of the anomalous process under the Feynman-Kac transform by the dual Feynman-Kac transform in terms of the weak dual process \(\widehat{X}_t\) and the inverse subordinator \(\{ E_t; t \geq 0\}\). We then establish the regularity of the density function, and show that it is the unique mild solution as well as the unique weak solution of a non-local Fokker-Planck equation that involves the dual generator of \(X\) and the potential measure of the subordinator \(S\). During the course of the study, we are naturally led to extend the notation of Riemann-Liouville integral to measures that are locally finite on \([0, \infty)\).Characterization of continuous symmetric distributions using information measures of recordshttps://www.zbmath.org/1483.620442022-05-16T20:40:13.078697Z"Ahmadi, Jafar"https://www.zbmath.org/authors/?q=ai:ahmadi.jafarSummary: In this paper, several characterizations of continuous symmetric distributions are provided. The results are based on the properties of some information measures of \(k\)-records. These include cumulative residual (past) entropy, Shannon entropy, Rényi entropy, Tsallis entropy, also some common Kerridge inaccuracy measures. It is proved that the equality of information in upper and lower \(k\)-records is a characteristic property of continuous symmetric distributions. Completeness properties of certain function sequences are also used to obtain some characterization results.Proposal for obtaining a priori distributions or the shape parameters of the beta distributionhttps://www.zbmath.org/1483.620452022-05-16T20:40:13.078697Z"Arroyo Bravo, Luis Gabriel"https://www.zbmath.org/authors/?q=ai:arroyo-bravo.luis-gabriel"Lasso Balanta, Fabian Alejandro"https://www.zbmath.org/authors/?q=ai:lasso-balanta.fabian-alejandro"Tovar Cuevas, José Rafael"https://www.zbmath.org/authors/?q=ai:tovar-cuevas.jose-rafaelSummary: The Beta distribution has gained importance in the field of applied statistics, because it is a probability function that allows us to model the natural behavior of random variables that can only take values within a delimited range of real numbers with finite length. A special case quite common in practice is that of the variables whose domain is within the interval \((0,1)\), the case of proportions or percentages. In this article, a methodology is proposed to obtain a priori distributions for the shape parameters of the Beta distribution, which entails a new approach for these distribution from the Bayesian paradigm, since, most of the time, it is used as a prior distribution when the sample data can be adjusted to a binomial distribution, but in the case of interest for this work, the sample data can be adjusted with a Beta distribution which would imply having a likelihood function that is the product of \(n\) densities Beta. A joint distribution for the distribution parameters was initially obtained from the information for the mean and the variance of the prior distribution. Finally, the a priori distributions for the shape parameters were obtained marginalizing the joint distribution.A regression model for positive data based on the slashed half-normal distributionhttps://www.zbmath.org/1483.620462022-05-16T20:40:13.078697Z"Gómez, Yolanda M."https://www.zbmath.org/authors/?q=ai:gomez.yolanda-m"Gallardo, Diego I."https://www.zbmath.org/authors/?q=ai:gallardo.diego-i"de Castro, Mário"https://www.zbmath.org/authors/?q=ai:de-castro.mario-hSummary: In this paper, we discuss several aspects about the slashed half-normal distribution. We reparameterize the model based on the mean and we perform comparisons with well-known regression models for positive data. Maximum likelihood estimation of the parameters is carried out through the expectation-maximization algorithm. Some properties of the estimators and two kinds of residuals are assessed in a simulation study. Two real datasets illustrated the proposed model as well as other three models for the sake of comparison.Flexible models for overdispersed and underdispersed count datahttps://www.zbmath.org/1483.620492022-05-16T20:40:13.078697Z"Cahoy, Dexter"https://www.zbmath.org/authors/?q=ai:cahoy.dexter-o"Di Nardo, Elvira"https://www.zbmath.org/authors/?q=ai:di-nardo.elvira"Polito, Federico"https://www.zbmath.org/authors/?q=ai:polito.federicoSummary: Within the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD's) to allow both overdispersion and underdispersion. Similarly to Kemp's generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD's related to a generalization of Mittag-Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.The length-biased Weibull-Rayleigh distribution for application to hydrological datahttps://www.zbmath.org/1483.620502022-05-16T20:40:13.078697Z"Chaito, Tanachot"https://www.zbmath.org/authors/?q=ai:chaito.tanachot"Khamkong, Manad"https://www.zbmath.org/authors/?q=ai:khamkong.manadSummary: In this study, we introduce a new class of length-biased distribution, the length-biased Weibull-Rayleigh (LBWR) distribution, and provide its properties such as the limit behavior, survival function, hazard rate function, \(r\)th moment, and moment generating function. Moreover, maximum likelihood estimation is used in the parameter estimation. The LBWR distribution was fitted to two hydrological datasets and its efficacy compared with Rayleigh, Weibull, Pareto, and Weibull-Rayleigh distributions, the results of which show that the novel distribution provides a better fit than the others.On the partial-geometric distribution: properties and applicationshttps://www.zbmath.org/1483.620522022-05-16T20:40:13.078697Z"Khruachalee, Krisada"https://www.zbmath.org/authors/?q=ai:khruachalee.krisada"Bodhisuwan, Winai"https://www.zbmath.org/authors/?q=ai:bodhisuwan.winai"Volodin, Andrei"https://www.zbmath.org/authors/?q=ai:volodin.andrei-iSummary: In this article we introduce the new, two-parameter partial-geometric distribution (PG) that contains both geometric and first success distributions as a particular case. Some probability and statistical properties of the proposed distribution are discussed, including probability mass function, mean, variance, moment generating function, and probability generating function. We propose the method of maximum likelihood for estimating the model's parameters, and apply the PG distribution to two real datasets to illustrate the flexibility of the proposed distribution. We found the PG distribution is more dynamic than the geometric distribution in the sense that it can be applied to the under-dispersed data. The PG distribution also performs well with a goodness of fit test and some other model selection characteristics for model fitting of these two datasets. Thus, the PG distribution can be applied as an alternative model for the analysis of discrete data.Asymptotic properties of likelihood ratio statistics in competing risks model under interval random censoringhttps://www.zbmath.org/1483.620592022-05-16T20:40:13.078697Z"Abdushukurov, A. A."https://www.zbmath.org/authors/?q=ai:abdushukurov.abdurahim-a|abdushukurov.abdurakhim-a"Nurmukhamedova, N. S."https://www.zbmath.org/authors/?q=ai:nurmukhamedova.n-sSummary: In this paper we consider local asymptotic normality of likelihood ratio statistics in competing risks model under random censoring by nonobserving intervals. Using the property of local asymptotic normality we investigate asymptotic properties of Bayesian type estimates for unknown parameter and prove its asymptotic efficiency.Almost sure convergence for END sequences and its application to \(M\) estimator in linear modelshttps://www.zbmath.org/1483.620612022-05-16T20:40:13.078697Z"Deng, Xin"https://www.zbmath.org/authors/?q=ai:deng.xin"Wang, Shijie"https://www.zbmath.org/authors/?q=ai:wang.shijie"Wang, Rui"https://www.zbmath.org/authors/?q=ai:wang.rui.2|wang.rui.1"Xie, Xiujuan"https://www.zbmath.org/authors/?q=ai:xie.xiujuan"Wang, Xuejun"https://www.zbmath.org/authors/?q=ai:wang.xuejunSummary: In the paper, an almost sure convergence result for weighted sums of extended negatively dependent random variables is obtained. By using the almost sure convergence result, we further study the strong consistency of \(M\) estimator of the regression parameter in linear models based on extended negatively dependent random errors under some mild conditions.Automated scalable Bayesian inference via Hilbert coresetshttps://www.zbmath.org/1483.620622022-05-16T20:40:13.078697Z"Campbell, Trevor"https://www.zbmath.org/authors/?q=ai:campbell.trevor"Broderick, Tamara"https://www.zbmath.org/authors/?q=ai:broderick.tamaraSummary: The automation of posterior inference in Bayesian data analysis has enabled experts and nonexperts alike to use more sophisticated models, engage in faster exploratory modeling and analysis, and ensure experimental reproducibility. However, standard automated posterior inference algorithms are not tractable at the scale of massive modern data sets, and modifications to make them so are typically model-specific, require expert tuning, and can break theoretical guarantees on inferential quality. Building on the Bayesian coresets framework, this work instead takes advantage of data redundancy to shrink the data set itself as a preprocessing step, providing fully-automated, scalable Bayesian inference with theoretical guarantees. We begin with an intuitive reformulation of Bayesian coreset construction as sparse vector sum approximation, and demonstrate that its automation and performance-based shortcomings arise from the use of the supremum norm. To address these shortcomings we develop Hilbert
coresets, i.e., Bayesian coresets constructed under a norm induced by an inner-product on the log-likelihood function space. We propose two Hilbert coreset construction algorithms -- one based on importance sampling, and one based on the Frank-Wolfe algorithm -- along with theoretical guarantees on approximation quality as a function of coreset size. Since the exact computation of the proposed inner-products is model-specific, we automate the construction with a random finite-dimensional projection of the log-likelihood functions. The resulting automated coreset construction algorithm is simple to implement, and experiments on a variety of models with real and synthetic data sets show that it provides high-quality posterior approximations and a significant reduction in the computational cost of inference.On a new method of the testing hypothesis of equality of two Bernoulli regression functions for group observationshttps://www.zbmath.org/1483.620742022-05-16T20:40:13.078697Z"Babilua, Petre"https://www.zbmath.org/authors/?q=ai:babilua.petre-k"Nadaraya, Elizbar"https://www.zbmath.org/authors/?q=ai:nadaraya.e-aSummary: In the paper, the limiting distribution is established for an integral square deviation of estimates of Bernoulli regression functions based on two group samples. Based on these results, the new test is constructed for the hypothesis testing on the equality of two Bernoulli regression functions. The question of consistency of the constructed test is studied, and the asymptotic of the test power is investigated for some close alternatives.Extremal behaviour of a periodically controlled sequence with imputed valueshttps://www.zbmath.org/1483.620852022-05-16T20:40:13.078697Z"Ferreira, Helena"https://www.zbmath.org/authors/?q=ai:ferreira.helena-maria-simoes|ferreira.helena"Martins, Ana Paula"https://www.zbmath.org/authors/?q=ai:martins.ana-paula"da Graça Temido, Maria"https://www.zbmath.org/authors/?q=ai:temido.m-gracaSummary: Extreme events are a major concern in statistical modeling. Random missing data can constitute a problem when modeling such rare events. Imputation is crucial in these situations and therefore models that describe different imputation functions enhance possible applications and enlarge the few known families of models that cover these situations. In this paper we consider a family of models \(\{Y_n\}\), \(n\ge 1,\) that can be associated to automatic systems which have a periodic control, in the sense that at instants multiple of \(T\), \(T\ge 2,\) no value is lost. Random missing values are here replaced by the biggest of the previous observations up to the one surely registered. We prove that when the underlying sequence is stationary, \(\{Y_n\}\) is \(T\)-periodic and, if it also verifies some local dependence conditions, then \(\{Y_n\}\) verifies one of the well known \(D^{(s)}_T(u_n)\), \(s\ge 1,\) dependence conditions for \(T\)-periodic sequences. We also obtain the extremal index of \(\{Y_n\}\) and relate it to the extremal index of the underlying sequence. A consistent estimator for the parameter that ``controls'' the missing values is here proposed and its finite sample properties are analysed. The obtained results are illustrated with Markovian sequences of recognized interest in applications.Multivariate tail covariance risk measure for generalized skew-elliptical distributionshttps://www.zbmath.org/1483.620862022-05-16T20:40:13.078697Z"Zuo, Baishuai"https://www.zbmath.org/authors/?q=ai:zuo.baishuai"Yin, Chuancun"https://www.zbmath.org/authors/?q=ai:yin.chuancunSummary: In this paper, the multivariate tail covariance (MTCov) for generalized skew-elliptical distributions is considered. Some special cases for this distribution, such as generalized skew-normal, generalized skew Student-\(t\), generalized skew-logistic and generalized skew-Laplace distributions, are also considered. In order to test the theoretical feasibility of our results, the MTCov for skewed and non skewed normal distributions is computed and compared. Finally, we give a special formula of the MTCov for generalized skew-elliptical distributions.On model selection for dense stochastic block modelshttps://www.zbmath.org/1483.620962022-05-16T20:40:13.078697Z"Norros, Ilkka"https://www.zbmath.org/authors/?q=ai:norros.ilkka"Reittu, Hannu"https://www.zbmath.org/authors/?q=ai:reittu.hannu"Bazsó, Fülöp"https://www.zbmath.org/authors/?q=ai:bazso.fulopSummary: This paper studies estimation of stochastic block models with Rissanen's minimum description length (MDL) principle in the dense graph asymptotics. We focus on the problem of model specification, i.e., identification of the number of blocks. Refinements of the true partition always decrease the code part corresponding to the edge placement, and thus a respective increase of the code part specifying the model should overweight that gain in order to yield a minimum at the true partition. The balance between these effects turns out to be delicate. We show that the MDL principle identifies the true partition among models whose relative block sizes are bounded away from zero. The results are extended to models with Poisson-distributed edge weights.Kernels for sequentially ordered datahttps://www.zbmath.org/1483.621442022-05-16T20:40:13.078697Z"Király, Franz J."https://www.zbmath.org/authors/?q=ai:kiraly.franz-j"Oberhauser, Harald"https://www.zbmath.org/authors/?q=ai:oberhauser.haraldSummary: We present a novel framework for learning with sequential data of any kind, such as multivariate time series, strings, or sequences of graphs. The main result is a ``sequentialization'' that transforms any kernel on a given domain into a kernel for sequences in that domain. This procedure preserves properties such as positive definiteness, the associated kernel feature map is an ordered variant of sample (cross-)moments, and this sequentialized kernel is consistent in the sense that it converges to a kernel for paths if sequences converge to paths (by discretization). Further, classical kernels for sequences arise as special cases of this method. We use dynamic programming and low-rank techniques for tensors to provide efficient algorithms to compute this sequentialized kernel.Change-level detection for Lévy subordinatorshttps://www.zbmath.org/1483.621462022-05-16T20:40:13.078697Z"Al Masry, Zeina"https://www.zbmath.org/authors/?q=ai:al-masry.zeina"Rabehasaina, Landy"https://www.zbmath.org/authors/?q=ai:rabehasaina.landy"Verdier, Ghislain"https://www.zbmath.org/authors/?q=ai:verdier.ghislainSummary: Let \(\boldsymbol{X} = (X_t)_{t \geq 0}\) be a process behaving as a general increasing Lévy process (subordinator) prior to hitting a given unknown level \(m_0\), then behaving as another different subordinator once this threshold is crossed. This paper addresses the detection of this unknown threshold \(m_0 \in [0, +\infty]\) from an observed trajectory of the process. These kind of model and issue are encountered in many areas such as reliability and quality control in degradation problems. More precisely, we construct, from a sample path and for each \(\epsilon > 0\), a so-called detection level \(L_{\epsilon}\) by considering a CUSUM inspired procedure. Under mild assumptions, this level is such that, while \(m_0\) is infinite (i.e. when no changes occur), its expectation \(\mathbb{E}_{\infty} (L_{\epsilon})\) tends to \(+\infty\) as \(\epsilon\) tends to 0, and the expected overshoot \(\mathbb{E}_{m_0} ([L_{\epsilon} -m_0]^+)\), while the threshold \(m_0\) is finite, is negligible compared to \(\mathbb{E}_{\infty} (L_{\epsilon})\) as \(\epsilon\) tends to 0. Numerical illustrations are provided when the Lévy processes are gamma processes with different shape parameters.Cluster point processes and Poisson thinning INARMAhttps://www.zbmath.org/1483.621492022-05-16T20:40:13.078697Z"Chen, Zezhun"https://www.zbmath.org/authors/?q=ai:chen.zezhun"Dassios, Angelos"https://www.zbmath.org/authors/?q=ai:dassios.angelosSummary: In this paper, we consider Poisson thinning Integer-valued time series models, namely integer-valued moving average model (INMA) and Integer-valued Autoregressive Moving Average model (INARMA), and their relationship with cluster point processes, the Cox point process and the dynamic contagion process. We derive the probability generating functionals of INARMA models and compare to that of cluster point processes. The main aim of this paper is to prove that, under a specific parametric setting, INMA and INARMA models are just discrete versions of continuous cluster point processes and hence converge weakly when the length of subintervals goes to zero.A note on estimation of \(\alpha\)-stable CARMA processes sampled at low frequencieshttps://www.zbmath.org/1483.621512022-05-16T20:40:13.078697Z"Fasen-Hartmann, Vicky"https://www.zbmath.org/authors/?q=ai:fasen-hartmann.vicky"Mayer, Celeste"https://www.zbmath.org/authors/?q=ai:mayer.celesteSummary: In this paper, we investigate estimators for \(\alpha\)-stable CARMA processes sampled equidistantly. Simulation studies suggest that the Whittle estimator and the estimator presented in [\textit{I. García} et al., Stat. Model. 11, No. 5, 447--470 (2011; Zbl 1420.62363)] are consistent estimators for the parameters of stable CARMA processes. For CARMA processes with finite second moments it is well-known that the Whittle estimator is consistent and asymptotically normally distributed. Therefore, in the light-tailed setting the properties of the Whittle estimator for CARMA processes are similar to those of the Whittle estimator for ARMA processes. However, in the present paper we prove that, in general, the Whittle estimator for \(\alpha\)-stable CARMA processes sampled at low frequencies is not consistent and highlight why simulation studies suggest something else. Thus, in contrast to the light-tailed setting the properties of the Whittle estimator for heavy-tailed ARMA processes cannot be transferred to heavy-tailed CARMA processes. We elaborate as well that the estimator presented in [loc. cit.] faces the same problems. However, the Whittle estimator for stable CAR(1) processes is consistent.Tracy-Widom law for the largest eigenvalue of sample covariance matrix generated by VARMAhttps://www.zbmath.org/1483.621552022-05-16T20:40:13.078697Z"Tian, Boping"https://www.zbmath.org/authors/?q=ai:tian.boping"Zhang, Yangchun"https://www.zbmath.org/authors/?q=ai:zhang.yangchun.1"Zhou, Wang"https://www.zbmath.org/authors/?q=ai:zhou.wangOn a time dependent divergence measure between two residual lifetime distributionshttps://www.zbmath.org/1483.621702022-05-16T20:40:13.078697Z"Mansourvar, Zahra"https://www.zbmath.org/authors/?q=ai:mansourvar.zahra"Asadi, Majid"https://www.zbmath.org/authors/?q=ai:asadi.majidSummary: Recently, a time-dependent measure of divergence has been introduced by Mansourvar and Asadi (2020) to assess the discrepancy between the survival functions of two residual lifetime random variables. In this paper, we derive various time-dependent results on the proposed divergence measure in connection to other well-known measures in reliability engineering. The proposed criterion is also examined in mixture models and a general class of survival transformation models which results in some well-known models in the lifetime studies and survival analysis. In addition, the time-dependent measure is employed to evaluate the divergence between the lifetime distributions of \(k\)-out-of-\(n\) systems and also to assess the discrepancy between the distribution functions of the epoch times of a non-homogeneous Poisson process.Selecting reduced models in the cross-entropy methodhttps://www.zbmath.org/1483.650112022-05-16T20:40:13.078697Z"Héas, P."https://www.zbmath.org/authors/?q=ai:heas.patrickFirst passage Monte Carlo algorithms for solving coupled systems of diffusion-reaction equationshttps://www.zbmath.org/1483.650132022-05-16T20:40:13.078697Z"Sabelfeld, Karl"https://www.zbmath.org/authors/?q=ai:sabelfeld.karl-kSummary: We suggest in this letter a new Random Walk on Spheres (RWS) stochastic algorithm for solving systems of coupled diffusion-reaction equations where the random walk is living both on the randomly walking spheres and inside the relevant balls. The method is mesh free both in space and time, and is well applied to solve high-dimensional problems with complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for balls and spheres. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain. Some applications to exciton flux calculations in the diffusion imaging method in semiconductors are discussed.Computing the density function of complex models with randomness by using polynomial expansions and the RVT technique. Application to the SIR epidemic modelhttps://www.zbmath.org/1483.650152022-05-16T20:40:13.078697Z"Calatayud, Julia"https://www.zbmath.org/authors/?q=ai:calatayud.julia"Carlos Cortés, Juan"https://www.zbmath.org/authors/?q=ai:cortes.juan-carlos"Jornet, Marc"https://www.zbmath.org/authors/?q=ai:jornet.marcSummary: This paper concerns the computation of the probability density function of the stochastic solution to general complex systems with uncertainties formulated via random differential equations. In the existing literature, the uncertainty quantification for random differential equations is based on the approximation of statistical moments by simulation or spectral methods, or on the computation of the exact density function via the random variable transformation (RVT) method when a closed-form solution is available. However, the problem of approximating the density function in a general setting has not been published yet. In this paper, we propose a hybrid method based on stochastic polynomial expansions, the RVT technique, and multidimensional integration schemes, to obtain accurate approximations to the solution density function. A problem-independent algorithm is proposed. The algorithm is tested on the SIR (susceptible-infected-recovered) epidemiological model, showing significant improvements compared to the previous literature.Development and application of the Fourier method for the numerical solution of Ito stochastic differential equationshttps://www.zbmath.org/1483.650162022-05-16T20:40:13.078697Z"Kuznetsov, D. F."https://www.zbmath.org/authors/?q=ai:kuznetsov.dmitriy-feliksovichSummary: This paper is devoted to the development and application of the Fourier method to the numerical solution of Ito stochastic differential equations. Fourier series are widely used in various fields of applied mathematics and physics. However, the method of Fourier series as applied to the numerical solution of stochastic differential equations, which are proper mathematical models of various dynamic systems affected by random disturbances, has not been adequately studied. This paper partially fills this gap.Exponential stability of \(\theta\)-EM method for nonlinear stochastic Volterra integro-differential equationshttps://www.zbmath.org/1483.650172022-05-16T20:40:13.078697Z"Lan, Guangqiang"https://www.zbmath.org/authors/?q=ai:lan.guangqiang"Zhao, Mei"https://www.zbmath.org/authors/?q=ai:zhao.mei"Qi, Siyuan"https://www.zbmath.org/authors/?q=ai:qi.siyuanSummary: Mean square exponential stability of both exact solutions and the corresponding \(\theta\)-EM method for stochastic Volterra integro-differential equations are investigated in this paper. For \(\frac{1}{2} < \theta \leq 1\), we prove that both exact solutions and the corresponding \(\theta\)-EM method for stochastic Volterra integro-differential equations are mean square exponentially stable under the Khasminskii-type conditions. If \(0 \leq \theta \leq \frac{1}{2}\), \(\theta\)-EM method is mean square exponentially stable under the Khasminskii-type condition plus linear growth condition on \(f\). By using Chebyshev inequality and Borel-Cantelli lemma, we can also prove that \(\theta\)-EM method is almost surely exponentially stable. An example is provided to support our conclusions.Convergence and stability of modified partially truncated Euler-Maruyama method for nonlinear stochastic differential equations with Hölder continuous diffusion coefficienthttps://www.zbmath.org/1483.650182022-05-16T20:40:13.078697Z"Yang, Hongfu"https://www.zbmath.org/authors/?q=ai:yang.hongfu"Huang, Jianhua"https://www.zbmath.org/authors/?q=ai:huang.jianhuaSummary: Recently, \textit{H. Yang} et al. [J. Comput. Appl. Math. 366, Article ID 112379, 13 p. (2020; Zbl 07126160)] established the strong convergence of the truncated Euler-Maruyama (EM) approximation, that was first proposed by \textit{X. Mao} [J. Comput. Appl. Math. 290, 370--384 (2015; Zbl 1330.65016)], for one-dimensional stochastic differential equations with superlinearly growing drift and the Hölder continuous diffusion coefficients. However, there are some restrictions on the truncation functions and these restrictions sometimes might force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is to construct several new techniques of the partially truncated EM method to establish the optimal convergence rate in theory without these restrictions. The other aim is to study the stability of the partially truncated EM method. Finally, some simulations and examples are provided to support the theoretical results and demonstrate the validity of the approach.Physics information aided kriging using stochastic simulation modelshttps://www.zbmath.org/1483.650192022-05-16T20:40:13.078697Z"Yang, Xiu"https://www.zbmath.org/authors/?q=ai:yang.xiu"Tartakovsky, Guzel"https://www.zbmath.org/authors/?q=ai:tartakovsky.guzel"Tartakovsky, Alexandre M."https://www.zbmath.org/authors/?q=ai:tartakovsky.alexandre-mCubic B-spline approximation for linear stochastic integro-differential equation of fractional orderhttps://www.zbmath.org/1483.652212022-05-16T20:40:13.078697Z"Mirzaee, Farshid"https://www.zbmath.org/authors/?q=ai:mirzaee.farshid"Alipour, Sahar"https://www.zbmath.org/authors/?q=ai:alipour.saharSummary: In this paper, the cubic B-spline collocation method is used for solving the stochastic integro-differential equation of fractional order. we show that stochastic integro-differential equation of fractional order is equivalent to a modified stochastic integral equation. Then we apply the proposed method to obtain a numerical scheme of the modified stochastic integral equation. Using this method, the problem solving turns into a linear system solution of equations. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.Parametric Markov chains: PCTL complexity and fraction-free Gaussian eliminationhttps://www.zbmath.org/1483.681952022-05-16T20:40:13.078697Z"Hutschenreiter, Lisa"https://www.zbmath.org/authors/?q=ai:hutschenreiter.lisa"Baier, Christel"https://www.zbmath.org/authors/?q=ai:baier.christel"Klein, Joachim"https://www.zbmath.org/authors/?q=ai:klein.joachimSummary: Parametric Markov chains have been introduced as a model for families of stochastic systems that rely on the same graph structure, but differ in the concrete transition probabilities. The latter are specified by polynomial constraints for the parameters. Among the tasks typically addressed in the analysis of parametric Markov chains are (1) the computation of closed-form solutions for reachabilty probabilities and other quantitative measures and (2) finding symbolic representations of the set of parameter valuations for which a given temporal logical formula holds as well as (3) the decision variant of (2) that asks whether there exists a parameter valuation where a temporal logical formula holds. Our contribution to (1) is to show that existing implementations for computing rational functions for reachability probabilities or expected costs in parametric Markov chains can be improved by using fraction-free Gaussian elimination, a long-known technique for linear equation systems with parametric coefficients. Our contribution to (2) and (3) is a complexity-theoretic discussion of the model checking problem for parametric Markov chains and probabilistic computation tree logic (PCTL) formulas. We present an exponential-time algorithm for (2) and a PSPACE upper bound for (3). Moreover, we identify fragments of PCTL and subclasses of parametric Markov chains where (1) and (3) are solvable in polynomial time and establish NP-hardness for other PCTL fragments.
For the entire collection see [Zbl 1436.68017].Bayesian inference by symbolic model checkinghttps://www.zbmath.org/1483.682042022-05-16T20:40:13.078697Z"Salmani, Bahare"https://www.zbmath.org/authors/?q=ai:salmani.bahare"Katoen, Joost-Pieter"https://www.zbmath.org/authors/?q=ai:katoen.joost-pieterSummary: This paper applies probabilistic model checking techniques for discrete Markov chains to inference in Bayesian networks. We present a simple translation from Bayesian networks into tree-like Markov chains such that inference can be reduced to computing reachability probabilities. Using a prototypical implementation on top of the Storm model checker, we show that symbolic data structures such as multi-terminal BDDs (MTBDDs) are very effective to perform inference on large Bayesian network benchmarks. We compare our result to inference using probabilistic sentential decision diagrams and vtrees, a scalable symbolic technique in AI inference tools.
For the entire collection see [Zbl 1475.68022].Incremental fuzzy probability decision-theoretic approaches to dynamic three-way approximationshttps://www.zbmath.org/1483.684152022-05-16T20:40:13.078697Z"Yang, Xin"https://www.zbmath.org/authors/?q=ai:yang.xin"Liu, Dun"https://www.zbmath.org/authors/?q=ai:liu.dun"Yang, Xibei"https://www.zbmath.org/authors/?q=ai:yang.xibei"Liu, Keyu"https://www.zbmath.org/authors/?q=ai:liu.keyu"Li, Tianrui"https://www.zbmath.org/authors/?q=ai:li.tianruiSummary: As a special model of three-way decision, three-way approximations in the fuzzy probability space can be interpreted, represented, and implemented as dividing the universe into three pair-wise disjoint regions, i.e., the positive, negative and boundary regions, which are transformed from the fuzzy membership grades with respect to the fuzzy concept. To consider the temporality and uncertainty of data simultaneously, this paper focuses on the integration of dynamics and fuzziness in the context of three-way approximations. We analyze and investigate three types of fuzzy conditional probability functions based on the fuzzy \(T\)-norm operators. Besides, we introduce the matrix-based fuzzy probability decision-theoretic models to dynamic three-way approximations based on the principle of least cost. Subsequently, to solve the time-consuming computational problem, we design the incremental algorithms by the updating strategies of matrices when the attributes evolve over time. Finally, a series of comparative experiments is reported to demonstrate and verify the performance of proposed models.Optimal approximations made easyhttps://www.zbmath.org/1483.685092022-05-16T20:40:13.078697Z"Csikós, Mónika"https://www.zbmath.org/authors/?q=ai:csikos.monika"Mustafa, Nabil H."https://www.zbmath.org/authors/?q=ai:mustafa.nabil-hassanSummary: The fundamental result of
\textit{Y. Li} et al. [J. Comput. Syst. Sci. 62, No. 3, 516--527 (2001; Zbl 0990.68081)]
on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, computational geometry, combinatorics, and data analysis.
The goal of this paper is to give a modular, self-contained, intuitive proof of this result for finite set systems. The only ingredient we assume is the standard Chernoff's concentration bound. This makes the proof accessible to a wider audience, readers not familiar with techniques from statistical learning theory, and makes it possible to be covered in a single self-contained lecture in a geometry, algorithms or combinatorics course.QBism and relational quantum mechanics comparedhttps://www.zbmath.org/1483.810092022-05-16T20:40:13.078697Z"Pienaar, Jacques"https://www.zbmath.org/authors/?q=ai:pienaar.jacquesSummary: The subjective Bayesian interpretation of quantum mechanics (QBism) and Rovelli's relational interpretation of quantum mechanics (RQM) are both notable for embracing the radical idea that measurement outcomes correspond to events whose occurrence (or not) is relative to an observer. Here we provide a detailed study of their similarities and especially their differences.Entanglement monotones connect distinguishability and predictabilityhttps://www.zbmath.org/1483.810222022-05-16T20:40:13.078697Z"Basso, Marcos L. W."https://www.zbmath.org/authors/?q=ai:basso.marcos-l-w"Maziero, Jonas"https://www.zbmath.org/authors/?q=ai:maziero.jonasSummary: Distinguishability and predictability appear in different complementarity relations. Englert and Bergou pointed out the possible connection among distinguishability, predictability, and entanglement. They conjectured that an entanglement measure was hidden between the measures of distinguishability and predictability. Qureshi connected these quantities for a particular trio of measures. In this letter, we define a new entropic distinguishability measure and suggest an entanglement measure as the difference between it and an entropic predictability measure from the literature. An entanglement monotone is defined from the largest value of the distinguishability and the corresponding predictability, provided that the predictability satisfies the criteria already established in the literature. Our results formally connect an entanglement monotone with distinguishability and the corresponding predictability, without appealing to specific measures.Noisy Simon period findinghttps://www.zbmath.org/1483.810442022-05-16T20:40:13.078697Z"May, Alexander"https://www.zbmath.org/authors/?q=ai:may.alexander"Schlieper, Lars"https://www.zbmath.org/authors/?q=ai:schlieper.lars"Schwinger, Jonathan"https://www.zbmath.org/authors/?q=ai:schwinger.jonathanSummary: Let \(f: \mathbb{F}_2^n \rightarrow \mathbb{F}_2^n\) be a Boolean function with period \(\mathbf{s} \). It is well-known that Simon's algorithm finds \(\mathbf{s}\) in time polynomial in \(n\) on quantum devices that are capable of performing error-correction. However, today's quantum devices are inherently noisy, too limited for error correction, and Simon's algorithm is not error-tolerant. We show that even noisy quantum period finding computations may lead to speedups in comparison to purely classical computations. To this end, we implemented Simon's quantum period finding circuit on the 15-qubit quantum device IBM Q 16 Melbourne. Our experiments show that with a certain probability \(\tau (n)\) we measure erroneous vectors that are not orthogonal to \(\mathbf{s} \). We propose new, simple, but very effective smoothing techniques to classically mitigate physical noise effects such as e.g. IBM Q's bias towards the 0-qubit.
After smoothing, our noisy quantum device provides us a statistical distribution that we can easily transform into an LPN instance with parameters \(n\) and \(\tau (n)\). Hence, in the noisy case we may not hope to find periods in time polynomial in \(n\). However, we may still obtain a quantum advantage if the error \(\tau (n)\) does not grow too large. This demonstrates that quantum devices may be useful for period finding, even before achieving the level of full error correction capability.
For the entire collection see [Zbl 1476.94005].Quantum windowed Fourier transform and its application to quantum signal processinghttps://www.zbmath.org/1483.810472022-05-16T20:40:13.078697Z"Yin, Haiting"https://www.zbmath.org/authors/?q=ai:yin.haiting"Lu, Dayong"https://www.zbmath.org/authors/?q=ai:lu.dayong"Zhang, Rui"https://www.zbmath.org/authors/?q=ai:zhang.rui.5|zhang.rui|zhang.rui.4|zhang.rui.2|zhang.rui.3|zhang.rui.1Summary: In classical information processing, the windowed Fourier transform (WFT), or short-time Fourier transform, which is a variant of the Fourier transform by dividing a longer time signal into shorter segments of equal length and then computing the Fourier transform separately on each shorter segment, is proposed to provide a method of signal processing. Up to now the discrete Fourier transform has been successfully applied to the field of quantum information, but the related short-time discrete Fourier transform of this field has not been developed accordingly. To address this problem, we first introduce the concept of quantum window state, and further prove that the quantum Fourier transform of a quantum window state is also a quantum window state. Based on the definition of the quantum window state the local information of a quantum signal is extracted and the corresponding quantum circuits are also given. And then, by applying the quantum Fourier transform to the windowed quantum superposition states, we propose a novel concept called quantum windowed Fourier transform (QWFT). Finally, an application of quantum signal processing is given where QWFT is employed.Autocorrelation functions for quantum particles in supersymmetric Pöschl-Teller potentialshttps://www.zbmath.org/1483.810802022-05-16T20:40:13.078697Z"Cellarosi, Francesco"https://www.zbmath.org/authors/?q=ai:cellarosi.francescoSummary: We consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric Pöschl-Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fernández provides large classes of Pöschl-Teller partner potentials to which our analysis applies.Two-dimensional quantum Yang-Mills theory and the Makeenko-Migdal equationshttps://www.zbmath.org/1483.811052022-05-16T20:40:13.078697Z"Lévy, Thierry"https://www.zbmath.org/authors/?q=ai:levy.thierrySummary: These notes, echoing a conference given at the Strasbourg-Zurich seminar in October 2017, are written to serve as an introduction to 2-dimensional quantum Yang-Mills theory and to the results obtained in the last five to ten years about its so-called large \(N\) limit.
For the entire collection see [Zbl 1473.53004].Generalized eigenfunctions for quantum walks via path counting approachhttps://www.zbmath.org/1483.811342022-05-16T20:40:13.078697Z"Komatsu, Takashi"https://www.zbmath.org/authors/?q=ai:komatsu.takashi"Konno, Norio"https://www.zbmath.org/authors/?q=ai:konno.norio"Morioka, Hisashi"https://www.zbmath.org/authors/?q=ai:morioka.hisashi"Segawa, Etsuo"https://www.zbmath.org/authors/?q=ai:segawa.etsuoStrongly coupled heavy and light quark thermal motion from AdS/CFThttps://www.zbmath.org/1483.811482022-05-16T20:40:13.078697Z"Mes, A. K."https://www.zbmath.org/authors/?q=ai:mes.a-k"Moerman, R. W."https://www.zbmath.org/authors/?q=ai:moerman.r-w"Shock, J. P."https://www.zbmath.org/authors/?q=ai:shock.jonathan-p"Horowitz, W. A."https://www.zbmath.org/authors/?q=ai:horowitz.w-aSummary: We give a pedagogical presentation of heavy and light probe quarks in a thermal plasma using the AdS/CFT correspondence. Three cases are considered: external heavy and light quarks undergoing Brownian motion in the plasma, and an external heavy quark moving through the plasma with a constant velocity. For the first two cases we compute the mean-squared transverse displacement of the string's boundary endpoint. At early times the behaviour is ballistic, while the late time dynamics are diffusive. We extract the diffusion coefficient for both heavy and light quarks in an arbitrary number of dimensions and comment on the relevance for relativistic heavy ion phenomenology.A model of spontaneous collapse with energy conservationhttps://www.zbmath.org/1483.811782022-05-16T20:40:13.078697Z"Snoke, D. W."https://www.zbmath.org/authors/?q=ai:snoke.david-wSummary: A model of spontaneous collapse of fermionic degrees of freedom in a quantum field is presented which has the advantages that it explicitly maintains energy conservation and gives results in agreement with an existing numerical method for calculating quantum state evolution, namely the quantum trajectories model.Noise-induced dynamics in a Josephson junction driven by trichotomous noiseshttps://www.zbmath.org/1483.820082022-05-16T20:40:13.078697Z"Jin, Yanfei"https://www.zbmath.org/authors/?q=ai:jin.yanfei"Wang, Heqiang"https://www.zbmath.org/authors/?q=ai:wang.heqiangSummary: Noise-induced dynamics is explored in a Josephson junction system driven by multiplicative and additive trichotomous noises in this paper. Under the adiabatic approximation, the analytical expression of average output current for the Josephson junction is obtained, which can be used to characterize stochastic resonance (SR). If only the additive trichotomous noise is considered, the large correlation time of additive noise can induce the suppression and the SR in the curve of average output current. When the effects of both multiplicative and additive trichotomous noises are considered, two pronounced peaks exist in the curves of average output current for large multiplicative noise amplitude and optimal additive noise intensity. That is, the stochastic multi-resonance phenomenon is observed in this system. Moreover, the curve of average output current appears a single peak as a function of multiplicative noise intensity, which disappears for the case of small fixed additive noise amplitude. Especially, the mean first-passage time (MFPT) as the function of additive trichotomous noise intensity displays a non-monotonic behavior with a maximum for the large multiplicative noise amplitude, which is called the phenomenon of the noise enhanced stability (NES).Constraint on the equation of state parameter (\(\omega\)) in non-minimally coupled \(f(Q)\) gravityhttps://www.zbmath.org/1483.830712022-05-16T20:40:13.078697Z"Mandal, Sanjay"https://www.zbmath.org/authors/?q=ai:mandal.sanjay-kumar"Sahoo, P. K."https://www.zbmath.org/authors/?q=ai:sahoo.pradyumn-kumarSummary: We study observational constraints on the modified symmetric teleparallel gravity, the non-metricity \(f(Q)\) gravity, which reproduces background expansion of the universe. For this purpose, we use Hubble measurements, Baryonic Acoustic Oscillations (BAO), 1048 Pantheon supernovae type Ia data sample which integrate SuperNova Legacy Survey (SNLS), Sloan Digital Sky Survey (SDSS), Hubble Space Telescope (HST) survey, Panoramic Survey Telescope and Rapid Response System (Pan-STARRS1). We confront our cosmological model against observational samples to set constraints on the parameters using Markov Chain Monte Carlo (MCMC) methods. We find the equation of state parameter \(\omega = -0.853_{-0.020}^{+0.015}\) and \(\omega = -0.796_{-0.074}^{+0.049}\) for Hubble and Pantheon samples, respectively. As a result, the \(f(Q)\) model shows the quintessence behavior and deviates from \(\Lambda\)CDM.Interpretation of algebraic inequalities. Practical engineering optimisation and generating new knowledgehttps://www.zbmath.org/1483.900082022-05-16T20:40:13.078697Z"Todinov, Michael"https://www.zbmath.org/authors/?q=ai:todinov.michael-tPublisher's description: This book introduces a new method based on algebraic inequalities for optimising engineering systems and processes, with applications in mechanical engineering, materials science, electrical engineering, reliability engineering, risk management and operational research.
This book shows that the application potential of algebraic inequalities in engineering and technology is far-reaching and certainly not restricted to specifying design constraints. Algebraic inequalities can handle deep uncertainty associated with design variables and control parameters. With the method presented in this book, powerful new knowledge about systems and processes can be generated through meaningful interpretation of algebraic inequalities. This book demonstrates how the generated knowledge can be put into practice through covering the algebraic inequalities suitable for interpretation in different contexts and describing how to apply this knowledge to enhance system and process performance. Depending on the specific interpretation, knowledge, applicable to different systems from different application domains, can be generated from the same algebraic inequality. Furthermore, an important class of algebraic inequalities has been introduced that can be used for optimising systems and processes in any area of science and technology provided that the variables and the separate terms of the inequalities are additive quantities.
With the presented various examples and solutions, this book will be of interest to engineers, students and researchers in the field of optimisation, engineering design, reliability engineering, risk management and operational research.A \(2\times 2\) random switching model and its dual risk modelhttps://www.zbmath.org/1483.900432022-05-16T20:40:13.078697Z"Behme, Anita"https://www.zbmath.org/authors/?q=ai:behme.anita-diana"Strietzel, Philipp Lukas"https://www.zbmath.org/authors/?q=ai:strietzel.philipp-lukasSummary: In this article, a special case of two coupled \(M/G/1\)-queues is considered, where two servers are exposed to two types of jobs that are distributed among the servers via a random switch. In this model, the asymptotic behavior of the workload buffer exceedance probabilities for the two single servers/both servers together/one (unspecified) server is determined. Hereby, one has to distinguish between jobs that are either heavy-tailed or light-tailed. The results are derived via the dual risk model of the studied coupled \(M/G/1\)-queues for which the asymptotic behavior of different ruin probabilities is determined.Matrix-geometric method for the analysis of a queuing system with perishable inventoryhttps://www.zbmath.org/1483.900442022-05-16T20:40:13.078697Z"Melikov, A. Z."https://www.zbmath.org/authors/?q=ai:melikov.agassi-z"Shahmaliyev, M. O."https://www.zbmath.org/authors/?q=ai:shahmaliyev.m-o"Nair, S. S."https://www.zbmath.org/authors/?q=ai:nair.satish-s|nair.sumitra-s|nair.sreejith-s|nair.sujath-s|nair.skreekantan-s|nair.sajeev-s|nair.seema-s|nair.swapna-s|nair.salini-sSummary: Markov models of queuing systems with perishable stocks and an infinite buffer are studied using two replenishment policies. In one of them, the volume of orders is constant, while the other depends on the current level of stocks. Customers can join the queue even when the inventory level is zero. After the service is completed, customers either receive supplies or leave the system without receiving them, while the duration of their service depends on whether the customer has received supplies or not. The conditions for the ergodicity of the constructed two-dimensional Markov chains are obtained, and the matrix-geometric method is used to calculate their steady-state distributions. Formulas are found for finding the characteristics of the system using the indicated replenishment policies, and the results of numerical experiments are given.A vector judgment on the MRL ordering for parallel systems with two exponential componentshttps://www.zbmath.org/1483.900452022-05-16T20:40:13.078697Z"Wang, Jiantian"https://www.zbmath.org/authors/?q=ai:wang.jiantianSummary: In this paper, we give a simple vector judgment on the MRL ordering of parallel systems with two independent exponential components.Pure Nash equilibria and best-response dynamics in random gameshttps://www.zbmath.org/1483.910282022-05-16T20:40:13.078697Z"Amiet, Ben"https://www.zbmath.org/authors/?q=ai:amiet.ben"Collevecchio, Andrea"https://www.zbmath.org/authors/?q=ai:collevecchio.andrea"Scarsini, Marco"https://www.zbmath.org/authors/?q=ai:scarsini.marco"Zhong, Ziwen"https://www.zbmath.org/authors/?q=ai:zhong.ziwenSummary: In finite games, mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies and the payoffs are independent and identically distributed with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of pure Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a pure Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that various phase transitions depend only on a single parameter of the model, that is, the probability of having ties.Controlled diffusion mean field games with common noise and McKean-Vlasov second order backward SDEshttps://www.zbmath.org/1483.910292022-05-16T20:40:13.078697Z"Barrasso, A."https://www.zbmath.org/authors/?q=ai:barrasso.adrien"Touzi, N."https://www.zbmath.org/authors/?q=ai:touzi.nizarControl and optimal stopping mean field games: a linear programming approachhttps://www.zbmath.org/1483.910322022-05-16T20:40:13.078697Z"Dumitrescu, Roxana"https://www.zbmath.org/authors/?q=ai:dumitrescu.roxana"Leutscher, Marcos"https://www.zbmath.org/authors/?q=ai:leutscher.marcos"Tankov, Peter"https://www.zbmath.org/authors/?q=ai:tankov.peterSummary: We develop the \textit{linear programming approach} to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the \textit{representative agent} chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in earlier papers in the pure control case.Extension of monotonic functions and representation of preferenceshttps://www.zbmath.org/1483.910812022-05-16T20:40:13.078697Z"Evren, Özgür"https://www.zbmath.org/authors/?q=ai:evren.ozgur"Hüsseinov, Farhad"https://www.zbmath.org/authors/?q=ai:husseinov.farhadSummary: Consider a dominance relation (a preorder) \( \succsim\) on a topological space \(X\), such as the \textit{greater than or equal to} relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set \(K \subseteq X\), we study when a continuous real function on \(K\) that is strictly monotonic with respect to \(\succsim\) can be extended to \(X\) without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when \(X\) is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space \(X\) starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.Posted price mechanisms and optimal threshold strategies for random arrivalshttps://www.zbmath.org/1483.910872022-05-16T20:40:13.078697Z"Correa, José"https://www.zbmath.org/authors/?q=ai:correa.jose-r"Foncea, Patricio"https://www.zbmath.org/authors/?q=ai:foncea.patricio"Hoeksma, Ruben"https://www.zbmath.org/authors/?q=ai:hoeksma.ruben"Oosterwijk, Tim"https://www.zbmath.org/authors/?q=ai:oosterwijk.tim"Vredeveld, Tjark"https://www.zbmath.org/authors/?q=ai:vredeveld.tjarkThe authors do present a mechanism design scheme between two parts, which is used for some time-dependent claim pricing. The incentive compatibility of this scheme is completely presented, such that the optimality of it is achieved. The pricing of the underlying claim is actually a partially equilibrium pricing, because one of the parts faces the claim as a time-dependent loss.
Reviewer: Christos E. Kountzakis (Karlovassi)Transitions in consumption behaviors in a peer-driven stochastic consumer networkhttps://www.zbmath.org/1483.911202022-05-16T20:40:13.078697Z"Jungeilges, Jochen"https://www.zbmath.org/authors/?q=ai:jungeilges.jochen-a"Ryazanova, Tatyana"https://www.zbmath.org/authors/?q=ai:ryazanova.tatyana-vladimirovnaSummary: We study transition phenomena between attractors occurring in a stochastic network of two consumers. The consumption of each individual is strongly influenced by the past consumption of the other individual, while own consumption experience only plays a marginal role. From a formal point of view we are dealing with a special case of a nonlinear stochastic consumption model taking the form of a 2-dimensional non-invertible map augmented by additive and/or parametric noise. In our investigation of the stochastic transitions we rely on a mixture of analytical and numerical techniques with a central role given to the concept of the stochastic sensitivity function and the related technique of confidence domains. We find that in the case of parametric noise the stochastic sensitivity of fixed points and cycles considered is considerably higher than in the case of additive noise. Three types of noise induced transitions between attractors are identified: (i) Escape from a stochastic fixed point with converge to a stochastic \(k\)-cycle, (ii) escape from the stochastic \(k\)-cycle to a stochastic fixed point, and (iii) cases in which the consumption process moves between the respective stochastic attractors for ever. The noise intensities at which such transitions are likely to occur tend to be smaller in the case of parametric noise than with additive noise.Steady states of lattice population models with immigrationhttps://www.zbmath.org/1483.911502022-05-16T20:40:13.078697Z"Chernousova, Elena"https://www.zbmath.org/authors/?q=ai:chernousova.elena"Feng, Yaqin"https://www.zbmath.org/authors/?q=ai:feng.yaqin"Hryniv, Ostap"https://www.zbmath.org/authors/?q=ai:hryniv.ostap"Molchanov, Stanislav"https://www.zbmath.org/authors/?q=ai:molchanov.stanislav-alekseevich"Whitmeyer, Joseph"https://www.zbmath.org/authors/?q=ai:whitmeyer.joseph-mSummary: In a lattice population model where individuals evolve as subcritical branching random walks subject to external immigration, the cumulants are estimated and the existence of the steady state is proved. The resulting dynamics are Lyapunov stable in that their qualitative behavior does not change under suitable perturbations of the main parameters of the model. An explicit formula of the limit distribution is derived in the solvable case of no birth. Monte Carlo simulation shows the limit distribution in the solvable case.Iterated learning in dynamic social networkshttps://www.zbmath.org/1483.911572022-05-16T20:40:13.078697Z"Chazelle, Bernard"https://www.zbmath.org/authors/?q=ai:chazelle.bernard"Wang, Chu"https://www.zbmath.org/authors/?q=ai:wang.chuSummary: A classic finding by \textit{M. L Kalish} et al. [``Iterated learning: intergenerational knowledge transmission reveals inductive biases'', Psychon. Bulletin Rev. 14, No. 2, 288--294 (2007; \url{doi:10.3758/bf03194066})] shows that no language can be learned iteratively by rational agents in a self-sustained manner. In other words, if \(A\) teaches a foreign language to \(B\), who then teaches what she learned to \(C\), and so on, the language will quickly get lost and agents will wind up teaching their own common native language. If so, how can linguistic novelty ever be sustained? We address this apparent paradox by considering the case of iterated learning in a social network: we show that by varying the lengths of the learning sessions over time or by keeping the networks dynamic, it is possible for iterated learning to endure forever with arbitrarily small loss.Minimizing the probability of lifetime ruin with deferred life annuitieshttps://www.zbmath.org/1483.911782022-05-16T20:40:13.078697Z"Bayraktar, Erhan"https://www.zbmath.org/authors/?q=ai:bayraktar.erhan"Young, Virginia R."https://www.zbmath.org/authors/?q=ai:young.virginia-rSummary: We find the minimum probability of lifetime ruin of an investor who can invest in a market with a risky and a riskless asset and who can purchase a deferred life annuity. Although we let the admissible set of strategies of annuity purchasing process be the set of increasing adapted processes, we find that the individual will not buy a deferred life annuity unless she can cover all her consumption via the annuity and have enough wealth left over to sustain her until the end of the deferral period.Analysis of a generalized penalty function in a semi-Markovian risk modelhttps://www.zbmath.org/1483.911822022-05-16T20:40:13.078697Z"Cheung, Eric C. K."https://www.zbmath.org/authors/?q=ai:cheung.eric-c-k"Landriault, David"https://www.zbmath.org/authors/?q=ai:landriault.davidSummary: In this paper an extension of the semi-Markovian risk model studied by \textit{H. Albrecher} and \textit{O. J. Boxma} [Insur. Math. Econ. 37, No. 3, 650--672 (2005; Zbl 1129.91023)] is considered by allowing for general interclaim times. In such a model, we follow the ideas of \textit{E. C. K. Cheung} et al. [Insur. Math. Econ. 46, No. 1, 117--126 (2010; Zbl 1231.91157)] and consider a generalization of the Gerber-Shiu function by incorporating two more random variables in the traditional penalty function, namely, the minimum surplus level before ruin and the surplus level immediately after the second last claim prior to ruin. It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation. Detailed examples are also considered when either the interclaim times or the claim sizes are exponentially distributed. Finally, we also consider the case where the claim arrival process follows a Markovian arrival process. Probabilistic arguments are used to derive the discounted joint distribution of four random variables of interest in this risk model by capitalizing on an existing connection with a particular fluid flow process.Note on stability of the ruin time density in a Sparre Andersen risk model with exponential claim sizeshttps://www.zbmath.org/1483.911962022-05-16T20:40:13.078697Z"Gordienko, E."https://www.zbmath.org/authors/?q=ai:gordienko.evgueni-i"De Chávez, J. Ruiz"https://www.zbmath.org/authors/?q=ai:ruiz-de-chavez.juan"Vázquez-Ortega, P."https://www.zbmath.org/authors/?q=ai:vazquez-ortega.patriciaSummary: In this note, the Sparre Andersen risk process with exponential claim sizes is considered. We derive upper bounds for deviations of the ruin time density when approximating the inter-claim time distribution. In particular, we treat approximation by means of empirical densities.The multi-dimensional stochastic Stefan financial model for a portfolio of assetshttps://www.zbmath.org/1483.912112022-05-16T20:40:13.078697Z"Antonopoulou, Dimitra C."https://www.zbmath.org/authors/?q=ai:antonopoulou.dimitra-c"Bitsaki, Marina"https://www.zbmath.org/authors/?q=ai:bitsaki.marina"Karali, Georgia"https://www.zbmath.org/authors/?q=ai:karali.georgia-dSummary: The financial model proposed involves the liquidation process of a portfolio through sell / buy orders placed at a price \(x\in\mathbb{R}^n\), with volatility. Its rigorous mathematical formulation results to an \(n\)-dimensional outer parabolic Stefan problem with noise. The moving boundary encloses the areas of zero trading. We will focus on a case of financial interest when one or more markets are considered. We estimate the areas of zero trading with diameter approximating the minimum of the \(n\) spreads for orders from the limit order books. In dimensions \(n = 3\), for zero volatility, this problem stands as a mean field model for Ostwald ripening, and has been proposed and analyzed by
\textit{B. Niethammer} in [Arch. Ration. Mech. Anal. 147, No. 2, 119--178 (1999; Zbl 0947.76092)],
and in [\textit{D. C. Antonopoulou} et al., J. Differ. Equations 252, No. 9, 4679--4718 (2012; Zbl 1244.35165)].
We propose a spherical moving boundaries approach where the zero trading area consists of a union of spherical domains centered at portfolios various prices with radii representing the half of the minimum spread. We apply Itô calculus and provide second order formal asymptotics for the stochastic dynamics of the spreads that seem to disconnect the financial model from a large diffusion assumption on the liquidity coefficient of the Laplacian that would correspond to an increased trading density. Moreover, we solve the approximating systems numerically.Sub- and supersolution approach to accuracy analysis of portfolio optimization asymptotics in multiscale stochastic factor marketshttps://www.zbmath.org/1483.912142022-05-16T20:40:13.078697Z"Fouque, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:fouque.jean-pierre"Hu, Ruimeng"https://www.zbmath.org/authors/?q=ai:hu.ruimeng"Sircar, Ronnie"https://www.zbmath.org/authors/?q=ai:sircar.ronnieA mean field game of optimal portfolio liquidationhttps://www.zbmath.org/1483.912152022-05-16T20:40:13.078697Z"Fu, Guanxing"https://www.zbmath.org/authors/?q=ai:fu.guanxing"Graewe, Paulwin"https://www.zbmath.org/authors/?q=ai:graewe.paulwin"Horst, Ulrich"https://www.zbmath.org/authors/?q=ai:horst.ulrich"Popier, Alexandre"https://www.zbmath.org/authors/?q=ai:popier.alexandreSummary: We consider a mean field game (MFG) of optimal portfolio liquidation under asymmetric information. We prove that the solution to the MFG can be characterized in terms of a forward-backward stochastic differential equation (FBSDE) with a possibly singular terminal condition on the backward component or, equivalently, in terms of an FBSDE with a finite terminal value yet a singular driver. Extending the method of continuation to linear-quadratic FBSDEs with a singular driver, we prove that the MFG has a unique solution. Our existence and uniqueness result allows proving that the MFG with a possibly singular terminal condition can be approximated by a sequence of MFGs with finite terminal values.A change-point approach for the identification of financial extreme regimeshttps://www.zbmath.org/1483.912222022-05-16T20:40:13.078697Z"Lattanzi, Chiara"https://www.zbmath.org/authors/?q=ai:lattanzi.chiara"Leonelli, Manuele"https://www.zbmath.org/authors/?q=ai:leonelli.manueleSummary: Inference over tails is usually performed by fitting an appropriate limiting distribution over observations that exceed a fixed threshold. However, the choice of such threshold is critical and can affect the inferential results. Extreme value mixture models have been defined to estimate the threshold using the full dataset and to give accurate tail estimates. Such models assume that the tail behavior is constant for all observations. However, the extreme behavior of financial returns often changes considerably in time and such changes occur by sudden shocks of the market. Here the extreme value mixture model class is extended to formally take into account distributional extreme change-points, by allowing for the presence of regime-dependent parameters modelling the tail of the distribution. This extension formally uses the full dataset to both estimate the thresholds and the extreme change-point locations, giving uncertainty measures for both quantities. Estimation of functions of interest in extreme value analyses is performed via MCMC algorithms. Our approach is evaluated through a series of simulations, applied to real data sets and assessed against competing approaches. Evidence demonstrates that the inclusion of different extreme regimes outperforms both static and dynamic competing approaches in financial applications.Pricing some life-contingent lookback options under regime-switching Lévy modelshttps://www.zbmath.org/1483.912262022-05-16T20:40:13.078697Z"Ai, Meiqiao"https://www.zbmath.org/authors/?q=ai:ai.meiqiao"Zhang, Zhimin"https://www.zbmath.org/authors/?q=ai:zhang.zhimin|zhang.zhimin.1Summary: In this paper, we study the valuation problem of life-contingent lookback options embedded in variable annuity with guaranteed minimum death benefit (GMDB). Specifically, the underlying asset price process is assumed to be an exponential regime-switching Lévy process, which is observed periodically. The Fourier cosine series expansion method is applied to compute exponential moments of the discretely monitored maximum and minimum of the regime-switching Lévy process. Furthermore, some explicit pricing formulas for the life-contingent lookback options embedded in GMDB products are derived. Finally, numerical experiments confirm the accuracy and efficiency of our method.On smile properties of volatility derivatives: understanding the VIX skewhttps://www.zbmath.org/1483.912272022-05-16T20:40:13.078697Z"Alòs, Elisa"https://www.zbmath.org/authors/?q=ai:alos.elisa"García-Lorite, David"https://www.zbmath.org/authors/?q=ai:garcia-lorite.david"Gonzalez, Aitor Muguruza"https://www.zbmath.org/authors/?q=ai:gonzalez.aitor-muguruzaThe pricing and numerical analysis of lookback options for mixed fractional Brownian motionhttps://www.zbmath.org/1483.912302022-05-16T20:40:13.078697Z"Chen, Qisheng"https://www.zbmath.org/authors/?q=ai:chen.qisheng"Zhang, Qian"https://www.zbmath.org/authors/?q=ai:zhang.qian"Liu, Chuan"https://www.zbmath.org/authors/?q=ai:liu.chuanSummary: Using the stochastic differential equation driven by the composite Poisson process of mixed fractional Brownian motion, the price model of a mixed jump-diffusion fractional Brownian motion environment is established. Under the condition of Merton's assumption, the Cauchy initial value problem of stochastic differential equations is iterated. The method is estimated, and the Merton formula of the European put option under the mixed jump-diffusion model is obtained, and the call-back option and the bearish option pricing formula of the mixed jump-diffusion fractional Brownian motion European floating strike price are given.Computation of powered option prices under a general model for underlying asset dynamicshttps://www.zbmath.org/1483.912332022-05-16T20:40:13.078697Z"Kim, Jerim"https://www.zbmath.org/authors/?q=ai:kim.jerim"Kim, Bara"https://www.zbmath.org/authors/?q=ai:kim.bara"Kim, Jeongsim"https://www.zbmath.org/authors/?q=ai:kim.jeongsim"Lee, Sungji"https://www.zbmath.org/authors/?q=ai:lee.sungjiSummary: We derive the Laplace transforms for the prices and deltas of the powered call and put options, as well as for the price and delta of the capped powered call option under a general framework. These Laplace transforms are expressed in terms of the transform of the underlying asset price at maturity. For any model that can derive the transform of the underlying asset price, we can obtain the Laplace transforms for the prices and deltas of the powered options and the capped powered call option. The prices and deltas of the powered options and the capped powered call option can be computed by numerical inversion of the Laplace transforms. Models to which our method can be applied include the geometric Lévy model, the regime-switching model, the Black-Scholes-Vasiček model, and Heston's stochastic volatility model, which are commonly used for pricing of financial derivatives. In this paper, numerical examples are presented for all four models.Pricing of defaultable options with multiscale generalized Heston's stochastic volatilityhttps://www.zbmath.org/1483.912342022-05-16T20:40:13.078697Z"Lee, Min-Ku"https://www.zbmath.org/authors/?q=ai:lee.min-ku"Kim, Jeong-Hoon"https://www.zbmath.org/authors/?q=ai:kim.jeong-hoonSummary: The possibility of default risk of an option writer becomes a more important issue in over-the-counter option market when systemic risk increases. It is desirable for the option price to reflect the default risk. On the other hand, it is known that a single scale, single factor stochastic volatility model such as the well-known Heston model would not price correctly in- and out-of-the money options. So, this paper studies the pricing of defaultable options under a multiscale generalized Heston's stochastic volatility model introduced by \textit{J.-P. Fouque} and \textit{M. J. Lorig} [SIAM J. Financ. Math. 2, 221--254 (2011; Zbl 1217.91189)] to resolve these issues. We derive an explicit solution formula for the defaultable option price and investigate the characteristics of the resultant price in comparison to the price under the original Heston model.\(\alpha\)-hypergeometric uncertain volatility models and their connection to 2BSDEshttps://www.zbmath.org/1483.912392022-05-16T20:40:13.078697Z"Mezdoud, Zaineb"https://www.zbmath.org/authors/?q=ai:mezdoud.zaineb"Hartmann, Carsten"https://www.zbmath.org/authors/?q=ai:hartmann.carsten"Remita, Mohamed Riad"https://www.zbmath.org/authors/?q=ai:remita.mohamed-riad"Kebiri, Omar"https://www.zbmath.org/authors/?q=ai:kebiri.omarSummary: In this article we propose a \(\alpha\)-hypergeometric model with uncertain volatility (UV) where we derive a worst-case scenario for option pricing. The approach is based on the connection between a certain class of nonlinear partial differential equations of HJB-type (G-HJB equations), that governs the nonlinear expectation of the UV model and provides an alternative to the difficult model calibration problem of UV models, and second-order backward stochastic differential equations (2BSDEs). Using asymptotic analysis for the G-HJB equation and the equivalent 2BSDE representation, we derive a limit model which provides an accurate description of the worst-case price scenario in cases when the bounds of the UV model are slowly varying. The analytical results are tested by numerical simulations using a deep learning based approximation of the underlying 2BSDE.Valuation of discrete dynamic fund protection under Lévy processeshttps://www.zbmath.org/1483.912422022-05-16T20:40:13.078697Z"Wong, Hoi Ying"https://www.zbmath.org/authors/?q=ai:wong.hoi-ying"Lam, Ka Wai"https://www.zbmath.org/authors/?q=ai:lam.ka-waiSummary: This paper investigates the valuation of discrete dynamic fund protection (DFP) under Lévy processes. Specifically, the analytical solution of discrete DFP under Lévy processes is obtained in terms of Fourier transforms. The derivation uses Spitzer's formula and leads to a recursion on computing the characteristic function of the maximum protection-to-fund ratio using the Fourier inversion. DFP can then be valued efficiently and accurately via the fast Fourier transform. The pricing behavior of the discrete DFP is numerically examined using several Lévy processes, such as geometric Brownian motion, jump-diffusion models, and variance gamma process. Numerical experiments confirm that the proposed approach produces highly accurate discrete DFP values within 1 second.Discussion on: ``Valuation of discrete dynamic fund protection under Lévy processes''https://www.zbmath.org/1483.912432022-05-16T20:40:13.078697Z"Yang, Jun"https://www.zbmath.org/authors/?q=ai:yang.jun|yang.jun.1|yang.jun.2|yang.jun.3Summary: Discussion of the paper [\textit{H. Y. Wong} and \textit{K. W. Lam}, ibid. 13, No. 2, 202--216 (2009; \url{doi:10.1080/10920277.2009.10597548})].An adaptive and explicit fourth order Runge-Kutta-Fehlberg method coupled with compact finite differencing for pricing American put optionshttps://www.zbmath.org/1483.912572022-05-16T20:40:13.078697Z"Nwankwo, Chinonso"https://www.zbmath.org/authors/?q=ai:nwankwo.chinonso"Dai, Weizhong"https://www.zbmath.org/authors/?q=ai:dai.weizhongSummary: We propose an adaptive and explicit Runge-Kutta-Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. Furthermore, we implement an extrapolation method to ensure that at least, a third-order accuracy in space is maintained at the boundary point when computing the optimal exercise boundary from its derivative. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge-Kutta-Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in the numerical experiment, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution.Inhibition enhances the coherence in the Jacobi neuronal modelhttps://www.zbmath.org/1483.920362022-05-16T20:40:13.078697Z"D'Onofrio, Giuseppe"https://www.zbmath.org/authors/?q=ai:donofrio.giuseppe"Lansky, Petr"https://www.zbmath.org/authors/?q=ai:lansky.petr"Tamborrino, Massimiliano"https://www.zbmath.org/authors/?q=ai:tamborrino.massimilianoSummary: The output signal is examined for the Jacobi neuronal model which is characterized by input-dependent multiplicative noise. The dependence of the noise on the rate of inhibition turns out to be of primary importance to observe maxima both in the output firing rate and in the diffusion coefficient of the spike count and, simultaneously, a minimum in the coefficient of variation (Fano factor). Moreover, we observe that an increment of the rate of inhibition can increase the degree of coherence computed from the power spectrum. This means that inhibition can enhance the coherence and thus the information transmission between the input and the output in this neuronal model. Finally, we stress that the firing rate, the coefficient of variation and the diffusion coefficient of the spike count cannot be used as the only indicator of coherence resonance without considering the power spectrum.From univariate to multivariate coupling between continuous signals and point processes: a mathematical frameworkhttps://www.zbmath.org/1483.920392022-05-16T20:40:13.078697Z"Safavi, Shervin"https://www.zbmath.org/authors/?q=ai:safavi.shervin"Logothetis, Nikos K."https://www.zbmath.org/authors/?q=ai:logothetis.nikos-k"Besserve, Michel"https://www.zbmath.org/authors/?q=ai:besserve.michelSummary: Time series data sets often contain heterogeneous signals, composed of both continuously changing quantities and discretely occurring events. The coupling between these measurements may provide insights into key underlying mechanisms of the systems under study. To better extract this information, we investigate the asymptotic statistical properties of coupling measures between continuous signals and point processes. We first introduce martingale stochastic integration theory as a mathematical model for a family of statistical quantities that include the phase locking value, a classical coupling measure to characterize complex dynamics. Based on the martingale central limit theorem, we can then derive the asymptotic Gaussian distribution of estimates of such coupling measure that can be exploited for statistical testing. Second, based on multivariate extensions of this result and random matrix theory, we establish a principled way to analyze the low-rank coupling between a large number of point processes and continuous signals. For a null hypothesis of no coupling, we establish sufficient conditions for the empirical distribution of squared singular values of the matrix to converge, as the number of measured signals increases, to the well-known Marchenko-Pastur (MP) law, and the largest squared singular value converges to the upper end of the MP support. This justifies a simple thresholding approach to assess the significance of multivariate coupling. Finally, we illustrate with simulations the relevance of our univariate and multivariate results in the context of neural time series, addressing how to reliably quantify the interplay between multichannel local field potential signals and the spiking activity of a large population of neurons.A quantitative study on the role of TKI combined with Wnt/\(\beta\)-catenin signaling and IFN-\(\alpha\) in the treatment of CML through deterministic and stochastic approacheshttps://www.zbmath.org/1483.920672022-05-16T20:40:13.078697Z"Pan, Sonjoy"https://www.zbmath.org/authors/?q=ai:pan.sonjoy"Raha, Soumyendu"https://www.zbmath.org/authors/?q=ai:raha.soumyendu"Chakrabarty, Siddhartha P."https://www.zbmath.org/authors/?q=ai:chakrabarty.siddhartha-pSummary: We propose deterministic and stochastic models for studying the pharmacokinetics of chronic myeloid leukemia (CML), in the presence of CTL immune response, upon administration of IFN-\(\alpha\) (the traditional treatment for CML), TKI (the current frontline medication for CML) and Wnt/\(\beta\)-catenin signaling (the state-of-the art therapeutic breakthrough for CML). To the best of our knowledge, no mathematical model incorporating all these three therapeutic protocols are available in literature. The stability analysis of the system equilibria is undertaken in terms of a threshold parameter. Further, this work introduces a stochastic approach in the study of CML dynamics. The evolution of the dynamics for both the deterministic and the stochastic models are examined. This study addresses the question of how the dual therapy of TKI and Wnt/\(\beta\)-catenin signaling or triple combination of all three, offers potentially improved therapeutic responses, particularly in terms of reducing side effects of TKI or IFN-\(\alpha\). The probability of CML extinction/remission based on the level of CML stem cells at detection is also predicted.The expected degree distribution in transient duplication divergence modelshttps://www.zbmath.org/1483.920692022-05-16T20:40:13.078697Z"Barbour, Andrew D."https://www.zbmath.org/authors/?q=ai:barbour.andrew-d"Lo, Tiffany Y. Y."https://www.zbmath.org/authors/?q=ai:lo.tiffany-y-ySummary: We study the degree distribution of a randomly chosen vertex in a duplication-divergence graph, under a variety of different generalizations of the basic model of \textit{A. Bhan} et al. [``A duplication growth model of gene expression networks'', Bioinformatics 18, No. 11, 1486--1493 (2002; \url{doi:10.1093/bioinformatics/18.11.1486})] and \textit{A. Vázquez} et al. [``Modeling of protein interaction networks'', Complexus 1, No. 1, 38--44 (2003; \url{doi:10.1159/000067642})]. We pay particular attention to what happens when a non-trivial proportion of the vertices have large degrees, establishing a central limit theorem for the logarithm of the degree distribution. Our approach, as in [\textit{J. Jordan}, ALEA, Lat. Am. J. Probab. Math. Stat. 15, No. 2, 1431--1445 (2018; Zbl 1403.05144)] and [\textit{F. Hermann} and \textit{P. Pfaffelhuber}, ALEA, Lat. Am. J. Probab. Math. Stat. 18, No. 1, 325--347 (2021; Zbl 1460.05173)], relies heavily on the analysis of related birth-catastrophe processes, and couplings are used to show that a number of different formulations of the process have asymptotically similar expected degree distributions.Quasi-neutral evolution in populations under small demographic fluctuationshttps://www.zbmath.org/1483.921082022-05-16T20:40:13.078697Z"Balasekaran, Madhumitha"https://www.zbmath.org/authors/?q=ai:balasekaran.madhumitha"Johanis, Michal"https://www.zbmath.org/authors/?q=ai:johanis.michal"Rychtář, Jan"https://www.zbmath.org/authors/?q=ai:rychtar.jan"Taylor, Dewey"https://www.zbmath.org/authors/?q=ai:taylor.dewey-t"Zhu, Jackie"https://www.zbmath.org/authors/?q=ai:zhu.jackieSummary: We study an eco-evolutionary dynamics in finite populations of two haploid asexually reproducing allelic types. We focus on the quasi-neutral case when individual types differ only in their intrinsic birth and death rates but have the same expected lifetime reproductive output. We assume that the population size can fluctuate stochastically. We solve the Kolmogorov forward equation in the population whose size fluctuates only minimally and show that the fixation probability is decreasing with the increasing turnover rate. We also show that when the mutant's turnover is small enough, selection favors the mutant replacing residents. Similarly, when the turnover is high enough, selection opposes the replacement. This basic result has previously been demonstrated numerically for the contact process and shown analytically for the Moran process; the current paper extends this analysis to provide an analytical proof for the contact process. We also demonstrate numerically that our results extend for general fluctuating populations and beyond the quasi-neutral case.The impact of poaching and regime switching on the dynamics of single-species modelhttps://www.zbmath.org/1483.921092022-05-16T20:40:13.078697Z"Du, Pengcheng"https://www.zbmath.org/authors/?q=ai:du.pengcheng"Liao, Yunhua"https://www.zbmath.org/authors/?q=ai:liao.yunhuaSummary: It is widely recognized that the criminal act of poaching has brought tremendous damage to biodiversity. This paper employs a stochastic single-species model with regime switching to investigate the impact of poaching. We first carry out the survival analysis and obtain sufficient conditions for the extinction and persistence in mean of the single-species population. Then, we show that the model is positive recurrent by constructing suitable Lyapunov function. Finally, numerical simulations are carried out to support our theoretical results. It is found that: (i) As the intensity of poaching increases, the odds of being at risk of extinction increases for the single-species population. (ii) The regime switching can suppress the extinction of the single-species population. (iii) The white noise is detrimental to the survival of the single-species population. (iv) Increasing the criminal cost of poaching and establishing animal sanctuaries are important ways to protect biodiversity.On the basic reproduction number in semi-Markov switching networkshttps://www.zbmath.org/1483.921242022-05-16T20:40:13.078697Z"Cao, Xiaochun"https://www.zbmath.org/authors/?q=ai:cao.xiaochun"Jin, Zhen"https://www.zbmath.org/authors/?q=ai:jin.zhen"Liu, Guirong"https://www.zbmath.org/authors/?q=ai:liu.gui-rong.1"Li, Michael Y."https://www.zbmath.org/authors/?q=ai:li.michael-yiSummary: Basic reproduction number \(\mathcal{R}_0\) in network epidemic dynamics is studied in the case of stochastic regime-switching networks. For generality, the dependence between successive networks is considered to follow a continuous time semi-Markov chain. \(\mathcal{R}_0\) is the weighted average of the basic reproduction numbers of deterministic subnetworks. Its position with respect to 1 can determine epidemic persistence or extinction in theories and simulations.Analysis of a stochastic distributed delay epidemic model with relapse and Gamma distribution kernelhttps://www.zbmath.org/1483.921252022-05-16T20:40:13.078697Z"Caraballo, Tomás"https://www.zbmath.org/authors/?q=ai:caraballo.tomas"El Fatini, Mohamed"https://www.zbmath.org/authors/?q=ai:el-fatini.mohamed"El Khalifi, Mohamed"https://www.zbmath.org/authors/?q=ai:el-khalifi.mohamed"Gerlach, Richard"https://www.zbmath.org/authors/?q=ai:gerlach.richard-h"Pettersson, Roger"https://www.zbmath.org/authors/?q=ai:pettersson.rogerSummary: In this work, we investigate a stochastic epidemic model with relapse and distributed delay. First, we prove that our model possesses and unique global positive solution. Next, by means of the Lyapunov method, we determine some sufficient criteria for the extinction of the disease and its persistence. In addition, we establish the existence of a unique stationary distribution to our model. Finally, we provide some numerical simulations for the stochastic model to assist and show the applicability and efficiency of our results.Escaping stochastic extinction of mutant virus: temporal pattern of emergence of drug resistance within a hosthttps://www.zbmath.org/1483.921322022-05-16T20:40:13.078697Z"Hayashi, Rena"https://www.zbmath.org/authors/?q=ai:hayashi.rena"Iwami, Shingo"https://www.zbmath.org/authors/?q=ai:iwami.shingo"Iwasa, Yoh"https://www.zbmath.org/authors/?q=ai:iwasa.yohSummary: After infecting a host, a viral strain may increase rapidly within the body and produce mutants with a faster proliferation rate than the virus itself. However, most of the mutants become extinct because of the stochasticity caused by the small number of infected cells. In addition, the mean growth rate of a mutant strain decreases with time because the number of susceptible target cells is reduced by the original strain. In this study, we calculated the fraction of mutants that can escape stochastic extinction, based on a continuous-time branching process with a time-dependent growth rate. We analyzed two cases differing in the mode of viral transmission: (1) an infected cell transmits the virus through cell-to-cell contact with a susceptible target cell; (2) an infected cell releases numerous free viral particles that subsequently infect susceptible target cells. The chance for a mutant strain to be established decreases with time after infection of the original type, and it may oscillate before convergence at the stationary value. We then calculated the probability of escaping stochastic extinction for a drug-resistant mutant when a patient received an antiviral drug that suppressed the original strain. Combining the rate of mutant production from the original strain and the chance of escaping stochastic extinction, the number of emerging drug-resistant mutations may have two peaks: one soon after the infection of the original type and the second at the start of antiviral drug administration.Estimating local outbreak risks and the effects of non-pharmaceutical interventions in age-structured populations: SARS-CoV-2 as a case studyhttps://www.zbmath.org/1483.921392022-05-16T20:40:13.078697Z"Lovell-Read, Francesca A."https://www.zbmath.org/authors/?q=ai:lovell-read.francesca-a"Shen, Silvia"https://www.zbmath.org/authors/?q=ai:shen.silvia"Thompson, Robin N."https://www.zbmath.org/authors/?q=ai:thompson.robin-nSummary: During the COVID-19 pandemic, non-pharmaceutical interventions (NPIs) including school closures, workplace closures and social distancing policies have been employed worldwide to reduce transmission and prevent local outbreaks. However, transmission and the effectiveness of NPIs depend strongly on age-related factors including heterogeneities in contact patterns and pathophysiology. Here, using SARS-CoV-2 as a case study, we develop a branching process model for assessing the risk that an infectious case arriving in a new location will initiate a local outbreak, accounting for the age distribution of the host population. We show that the risk of a local outbreak depends on the age of the index case, and we explore the effects of NPIs targeting individuals of different ages. Social distancing policies that reduce contacts outside of schools and workplaces and target individuals of all ages are predicted to reduce local outbreak risks substantially, whereas school closures have a more limited impact. In the scenarios considered here, when different NPIs are used in combination the risk of local outbreaks can be eliminated. We also show that heightened surveillance of infectious individuals reduces the level of NPIs required to prevent local outbreaks, particularly if enhanced surveillance of symptomatic cases is combined with efforts to find and isolate nonsymptomatic infected individuals. Our results reflect real-world experience of the COVID-19 pandemic, during which combinations of intense NPIs have reduced transmission and the risk of local outbreaks. The general modelling framework that we present can be used to estimate local outbreak risks during future epidemics of a range of pathogens, accounting fully for age-related factors.The effect of environmental noise on threshold dynamics for a stochastic viral infection model with two modes of transmission and immune impairmenthttps://www.zbmath.org/1483.921422022-05-16T20:40:13.078697Z"Ma, Yuanlin"https://www.zbmath.org/authors/?q=ai:ma.yuanlin"Yu, Xingwang"https://www.zbmath.org/authors/?q=ai:yu.xingwangSummary: A stochastic viral infection model with two modes of transmission and immune impairment is proposed in this paper, which contains four variables; uninfected cells, infected cells, virus particles and cytotoxic T lymphocytes. We first investigate the exponential stability of the model, and further give sufficient conditions for the extinction and persistence of the disease. By constructing a suitable stochastic Lyapunov function, we then prove the existence of a unique ergodic stationary distribution of the model. More importantly, a stochastic threshold \(\mathcal{R}_0^W\) is presented, which plays the similar role as \(\mathcal{R}_0\) in determining the persistence and extinction of the infected cells. As an application of the method proposed in this paper, the existence of an ergodic stationary distribution and a stochastic positive periodic solution of the model under the influence of colored noise and seasonal fluctuations are studied respectively. Finally, we carry out some numerical simulations, showing that environmental noise may suppress the spread of disease.A mean-field limit of the particle swarmalator modelhttps://www.zbmath.org/1483.921652022-05-16T20:40:13.078697Z"Ha, Seung-Yeal"https://www.zbmath.org/authors/?q=ai:ha.seung-yeal"Jung, Jinwook"https://www.zbmath.org/authors/?q=ai:jung.jinwook"Kim, Jeongho"https://www.zbmath.org/authors/?q=ai:kim.jeongho"Park, Jinyeong"https://www.zbmath.org/authors/?q=ai:park.jinyeong"Zhang, Xiongtao"https://www.zbmath.org/authors/?q=ai:zhang.xiongtaoAuthors' abstract: We present a mean-field limit of the particle swarmalator model introduced in [\textit{K. P. O'Keeffe} et al., ``Oscillators that sync and swarm'', Nature Commun. 8, 1504 (2017)] with singular communication weights. For a mean-field limit, we employ a probabilistic approach for the propagation of molecular chaos and suitable cut-offs in singular terms, which results in the validation of the mean-field limit. We also provide a local-in-time well-posedness of strong and weak solutions to the derived kinetic swarmalator equation.
Reviewer: Jiu-Gang Dong (Dalian)Distributed consensus over Markovian packet loss channelshttps://www.zbmath.org/1483.930122022-05-16T20:40:13.078697Z"Xu, Liang"https://www.zbmath.org/authors/?q=ai:xu.liang"Mo, Yilin"https://www.zbmath.org/authors/?q=ai:mo.yilin"Xie, Lihua"https://www.zbmath.org/authors/?q=ai:xie.lihuaEditorial remark: No review copy delivered.Local controllability of single-input nonlinear systems based on deterministic Wiener processeshttps://www.zbmath.org/1483.930382022-05-16T20:40:13.078697Z"Nishimura, Yûki"https://www.zbmath.org/authors/?q=ai:nishimura.yuki"Tsubakino, Daisuke"https://www.zbmath.org/authors/?q=ai:tsubakino.daisukeEditorial remark: No review copy delivered.Model reduction of Markovian jump systems with uncertain probabilitieshttps://www.zbmath.org/1483.930472022-05-16T20:40:13.078697Z"Shen, Ying"https://www.zbmath.org/authors/?q=ai:shen.ying|shen.ying.1"Wu, Zheng-Guang"https://www.zbmath.org/authors/?q=ai:wu.zhengguang"Shi, Peng"https://www.zbmath.org/authors/?q=ai:shi.peng"Ahn, Choon Ki"https://www.zbmath.org/authors/?q=ai:ahn.choon-kiEditorial remark: No review copy delivered.State-space realizations and optimal smoothing for Gaussian generalized reciprocal processeshttps://www.zbmath.org/1483.930772022-05-16T20:40:13.078697Z"White, Langford B."https://www.zbmath.org/authors/?q=ai:white.langford-b"Carravetta, Francesco"https://www.zbmath.org/authors/?q=ai:carravetta.francescoEditorial remark: No review copy delivered.Markov chains with maximum return time entropy for robotic surveillancehttps://www.zbmath.org/1483.934312022-05-16T20:40:13.078697Z"Duan, Xiaoming"https://www.zbmath.org/authors/?q=ai:duan.xiaoming"George, Mishel"https://www.zbmath.org/authors/?q=ai:george.mishel"Bullo, Francesco"https://www.zbmath.org/authors/?q=ai:bullo.francescoEditorial remark: No review copy delivered.Sequential nonparametric algorithm for detecting time series breakdownhttps://www.zbmath.org/1483.936102022-05-16T20:40:13.078697Z"Chervova, A. A."https://www.zbmath.org/authors/?q=ai:chervova.a-a"Filaretov, G. F."https://www.zbmath.org/authors/?q=ai:filaretov.g-f.1"Bouchaala Zineddine"https://www.zbmath.org/authors/?q=ai:bouchaala-zineddine.Summary: We consider the problem of detecting spontaneous changes in the characteristics of a time series (a time series breakdown), when the breakdown manifests itself in the form of a stepwise change in the value of the location parameter (expectation or median) of the probability distribution function of the controlled process. It is proposed to solve this problem in real time applying a sequential nonparametric detection algorithm based on using the random walk theory mechanism. By computer simulation we investigate the probabilistic characteristics of the proposed algorithm and its efficiency in comparison with the well-known CUSUM-algorithm of parametric type. A procedure for synthesizing a control algorithm with the specified properties is given. It emphasizes the prospects of using such an algorithm in monitoring systems for various purposes, usually created in conditions of shortage of a priori information about the properties of the controlled process.Sliding mode control for nonlinear stochastic singular semi-Markov jump systemshttps://www.zbmath.org/1483.936122022-05-16T20:40:13.078697Z"Qi, Wenhai"https://www.zbmath.org/authors/?q=ai:qi.wenhai"Zong, Guangdeng"https://www.zbmath.org/authors/?q=ai:zong.guangdeng"Karimi, Hamid Reza"https://www.zbmath.org/authors/?q=ai:karimi.hamid-reza|karimi.hamidreza-rEditorial remark: No review copy delivered.An analytical method for the analysis of inhomogeneous continuous Markov processes with piecewise constant transition intensitieshttps://www.zbmath.org/1483.936132022-05-16T20:40:13.078697Z"Vytovtov, K. A."https://www.zbmath.org/authors/?q=ai:vytovtov.k-a"Barabanova, E. A."https://www.zbmath.org/authors/?q=ai:barabanova.e-aSummary: The article deals with an inhomogeneous Markov process with finitely many discrete states, continuous time, and piecewise constant transition intensities. For the first time, analytical expressions are presented that describe both the transient and steady-state modes of the random process. To solve this problem, the fundamental matrix of the Kolmogorov system of differential equations is found in closed form in terms of elementary functions. In addition, an inhomogeneous process with periodically varying transition intensities is considered. For this case, the conditions for the existence of a steady-state mode are presented. Results of numerical calculations are provided for processes without jumps, with jumps, and with periodic jumps in the transition intensities.Probabilistic tracking control of dissipated Hamiltonian systems excited by Gaussian white noiseshttps://www.zbmath.org/1483.936162022-05-16T20:40:13.078697Z"Yang, Ying"https://www.zbmath.org/authors/?q=ai:yang.ying"Wang, Yong"https://www.zbmath.org/authors/?q=ai:wang.yong.10|wang.yong.8|wang.yong.9|wang.yong.5|wang.yong.11|wang.yong.6|wang.yong.3|wang.yong.2|wang.yong.7|wang.yong.1|wang.yong"Huang, Zhilong"https://www.zbmath.org/authors/?q=ai:huang.zhilongSummary: This paper devotes to a feedback control strategy for nonlinear stochastic dynamical system to track a prespecified stationary response probability density. The system description and control design are conducted in \textit{Hamiltonian framework}, and the excitations are confined to Gaussian white noises. The control design consists of several successive steps: firstly, separating the control into conservative and dissipative components by physical intuition, and expanding two components as polynomials; secondly, deriving the low-dimensional averaged equations of controlled Hamiltonian by stochastic averaging, and obtaining the stationary probability density of controlled responses by solving the associated Fokker-Planck-Kolmogorov (FPK) equation; thirdly and finally, determining the polynomial coefficients by minimising the performance index which balances the tracking performance and control cost. Two examples, i.e. Duffing oscillator and frictional system are adopted to illustrate the application and efficacy of this control strategy to track Gaussian and non-Gaussian response probability density.Observed-mode-dependent state estimation of hidden semi-Markov jump linear systemshttps://www.zbmath.org/1483.936172022-05-16T20:40:13.078697Z"Cai, Bo"https://www.zbmath.org/authors/?q=ai:cai.bo"Zhang, Lixian"https://www.zbmath.org/authors/?q=ai:zhang.lixian"Shi, Yang"https://www.zbmath.org/authors/?q=ai:shi.yangEditorial remark: No review copy delivered.On estimation errors in optical communication and locationhttps://www.zbmath.org/1483.936202022-05-16T20:40:13.078697Z"Chernoyarov, O. V."https://www.zbmath.org/authors/?q=ai:chernoyarov.oleg-v"Dachian, S."https://www.zbmath.org/authors/?q=ai:dachian.serguei"Kutoyants, Yu. A."https://www.zbmath.org/authors/?q=ai:kutoyants.yury-a"Zyulkov, A. V."https://www.zbmath.org/authors/?q=ai:zyulkov.a-vSummary: We consider several problems of parameter estimation based on observations of inhomogeneous Poisson processes arising in various practical applications of optical communication and location. The intensity function of the observed process consists of a periodic signal depending on an unknown parameter and a constant noise intensity. The asymptotic behavior of maximum likelihood and Bayesian estimators in cases of phase and frequency modulation of signals is described. Particular attention is paid to signals of various regularity (smooth, continuous but nondifferentiable, and of change-point type). Numerical simulations illustrate the results presented. This paper is a survey of results on the behavior of estimators in cases of frequency and phase modulation of signals of various regularity.Semi-Markov jump linear systems with incomplete sojourn and transition information: analysis and synthesishttps://www.zbmath.org/1483.936892022-05-16T20:40:13.078697Z"Ning, Zepeng"https://www.zbmath.org/authors/?q=ai:ning.zepeng"Zhang, Lixian"https://www.zbmath.org/authors/?q=ai:zhang.lixian"Colaneri, Patrizio"https://www.zbmath.org/authors/?q=ai:colaneri.patrizioEditorial remark: No review copy delivered.Averaging principle and stability of hybrid stochastic fractional differential equations driven by Lévy noisehttps://www.zbmath.org/1483.936912022-05-16T20:40:13.078697Z"Shen, Guangjun"https://www.zbmath.org/authors/?q=ai:shen.guangjun"Xiao, Ruidong"https://www.zbmath.org/authors/?q=ai:xiao.ruidong"Yin, Xiuwei"https://www.zbmath.org/authors/?q=ai:yin.xiuweiSummary: Stability of stochastic differential equations driven by Lévy noise with Markovian switching has recently received a lot of attention. Different from the integer-order stochastic differential equations, stochastic fractional differential equations play a circular role in describing many practical processes and systems. In this paper, our aims are to study the averaging principle of the solution of hybrid stochastic fractional differential equations driven by Lévy noise under non-Lipschitz conditions which include classical Lipschitz conditions as special cases and propose several sufficient conditions for asymptotic stability in the \(p\) th moment of the solution. Two examples with numerical simulation are given to illustrate the obtained theory.Stability of stochastic differential equations driven by the time-changed Lévy process with impulsive effectshttps://www.zbmath.org/1483.936982022-05-16T20:40:13.078697Z"Yin, Xiuwei"https://www.zbmath.org/authors/?q=ai:yin.xiuwei"Xu, Wentao"https://www.zbmath.org/authors/?q=ai:xu.wentao"Shen, Guangjun"https://www.zbmath.org/authors/?q=ai:shen.guangjunSummary: The stability of nonlinear stochastic differential equations driven by time-changed Lévy process with impulsive effects is discussed in this paper. Some sufficient conditions are provided to guarantee the solutions to be stable in different senses. The stochastic perturbation is also investigated for some unstable time-changed differential equations with impulses. The efficiency of the proposed results is illustrated by some examples with numerical simulations.Faster Johnson-Lindenstrauss transforms via Kronecker productshttps://www.zbmath.org/1483.940172022-05-16T20:40:13.078697Z"Jin, Ruhui"https://www.zbmath.org/authors/?q=ai:jin.ruhui"Kolda, Tamara G."https://www.zbmath.org/authors/?q=ai:kolda.tamara-g"Ward, Rachel"https://www.zbmath.org/authors/?q=ai:ward.rachel-aSummary: The Kronecker product is an important matrix operation with a wide range of applications in signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a generalization of the fast Johnson-Lindenstrauss projection for embedding vectors with Kronecker product structure, the Kronecker fast Johnson-Lindenstrauss transform (KFJLT). The KFJLT reduces the embedding cost by an exponential factor of the standard fast Johnson-Lindenstrauss transform's cost when applied to vectors with Kronecker structure, by avoiding explicitly forming the full Kronecker products. We prove that this computational gain comes with only a small price in embedding power: consider a finite set of \(p\) points in a tensor product of \(d\) constituent Euclidean spaces \(\bigotimes_{k=d}^1\mathbb{R}^{n_k}\), and let \(N = \prod_{k=1}^dn_k\). With high probability, a random KFJLT matrix of dimension \(m \times N\) embeds the set of points up to multiplicative distortion (\(1\pm \varepsilon\)) provided \(m \gtrsim \varepsilon^{-2}\,\log^{2d-1}(p)\,\log N\). We conclude by describing a direct application of the KFJLT to the efficient solution of large-scale Kronecker-structured least squares problems for fitting the CP tensor decomposition.RBFNN-based minimum entropy filtering for a class of stochastic nonlinear systemshttps://www.zbmath.org/1483.940212022-05-16T20:40:13.078697Z"Yin, Xin"https://www.zbmath.org/authors/?q=ai:yin.xin"Zhang, Qichun"https://www.zbmath.org/authors/?q=ai:zhang.qichun"Wang, Hong"https://www.zbmath.org/authors/?q=ai:wang.hong.7"Ding, Zhengtao"https://www.zbmath.org/authors/?q=ai:ding.zhengtaoEditorial remark: No review copy delivered.