Recent zbMATH articles in MSC 60J65https://www.zbmath.org/atom/cc/60J652021-04-16T16:22:00+00:00WerkzeugLocal large deviation principle for Wiener process with random resetting.https://www.zbmath.org/1456.600662021-04-16T16:22:00+00:00"Logachov, A."https://www.zbmath.org/authors/?q=ai:logachov.artem-vasilevich|logachov.artem-v"Logachova, O."https://www.zbmath.org/authors/?q=ai:logachova.olga-m"Yambartsev, A."https://www.zbmath.org/authors/?q=ai:yambartsev.anatoly-a|yambartsev.anatoliThe one-dimensional KPZ equation and the Airy process.https://www.zbmath.org/1456.827112021-04-16T16:22:00+00:00"Prolhac, Sylvain"https://www.zbmath.org/authors/?q=ai:prolhac.sylvain"Spohn, Herbert"https://www.zbmath.org/authors/?q=ai:spohn.herbertExtracting non-Gaussian governing laws from data on mean exit time.https://www.zbmath.org/1456.370932021-04-16T16:22:00+00:00"Zhang, Yanxia"https://www.zbmath.org/authors/?q=ai:zhang.yanxia"Duan, Jinqiao"https://www.zbmath.org/authors/?q=ai:duan.jinqiao"Jin, Yanfei"https://www.zbmath.org/authors/?q=ai:jin.yanfei"Li, Yang"https://www.zbmath.org/authors/?q=ai:li.yang.5Summary: Motivated by the existing difficulties in establishing mathematical models and in observing state time series for some complex systems, especially for those driven by non-Gaussian Lévy motion, we devise a method for extracting non-Gaussian governing laws with observations only on the mean exit time. It is feasible to observe the mean exit time for certain complex systems. With such observations, we use a sparse regression technique in the least squares sense to obtain the approximated function expression of the mean exit time. Then, we learn the generator and further identify the governing stochastic differential equation by solving an inverse problem for a nonlocal partial differential equation and minimizing an error objective function. Finally, we verify the efficacy of the proposed method by three examples with the aid of the simulated data from the original systems. Results show that our method can apply to not only the stochastic dynamical systems driven by Gaussian Brownian motion but also those driven by non-Gaussian Lévy motion, including those systems with complex rational drift.
{\copyright 2020 American Institute of Physics}Intermediate-level crossings of a first-passage path.https://www.zbmath.org/1456.602042021-04-16T16:22:00+00:00"Bhat, Uttam"https://www.zbmath.org/authors/?q=ai:bhat.uttam"Redner, S."https://www.zbmath.org/authors/?q=ai:redner.sidneyArithmetic area for \(m\) planar Brownian paths.https://www.zbmath.org/1456.602082021-04-16T16:22:00+00:00"Desbois, Jean"https://www.zbmath.org/authors/?q=ai:desbois.jean"Ouvry, Stéphane"https://www.zbmath.org/authors/?q=ai:ouvry.stephaneOccupation times for single-file diffusion.https://www.zbmath.org/1456.826482021-04-16T16:22:00+00:00"Bénichou, Olivier"https://www.zbmath.org/authors/?q=ai:benichou.olivier"Desbois, Jean"https://www.zbmath.org/authors/?q=ai:desbois.jeanSome results on the Brownian meander with drift.https://www.zbmath.org/1456.600902021-04-16T16:22:00+00:00"Iafrate, F."https://www.zbmath.org/authors/?q=ai:iafrate.francesco"Orsingher, E."https://www.zbmath.org/authors/?q=ai:orsingher.enzoSummary: In this paper we study the drifted Brownian meander that is a Brownian motion starting from \(u\) and subject to the condition that \(\min_{ 0\le z\le t}B(z)> v\) with \(u > v \). The limiting process for \(u\downarrow v\) is analysed, and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribution of which coincides with the driftless case.Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient.https://www.zbmath.org/1456.602052021-04-16T16:22:00+00:00"Boyer, Denis"https://www.zbmath.org/authors/?q=ai:boyer.denis"Dean, David S."https://www.zbmath.org/authors/?q=ai:dean.david-s"Mejía-Monasterio, Carlos"https://www.zbmath.org/authors/?q=ai:mejia-monasterio.carlos"Oshanin, Gleb"https://www.zbmath.org/authors/?q=ai:oshanin.glebThe genealogy of extremal particles of branching Brownian motion.https://www.zbmath.org/1456.602302021-04-16T16:22:00+00:00"Kliem, Sandra"https://www.zbmath.org/authors/?q=ai:kliem.sandra-m"Saha, Kumarjit"https://www.zbmath.org/authors/?q=ai:saha.kumarjitContinuity of zero-hitting times of Bessel processes and welding homeomorphisms of \(\operatorname{SLE}_k\).https://www.zbmath.org/1456.300392021-04-16T16:22:00+00:00"Beliaev, Dmitry"https://www.zbmath.org/authors/?q=ai:beliaev.dmitri-b"Margarint, Vlad"https://www.zbmath.org/authors/?q=ai:margarint.vlad"Shekhar, Atul"https://www.zbmath.org/authors/?q=ai:shekhar.atulSummary: We consider a family of Bessel Processes that depend on the starting point \(x\) and dimension \(\delta\), but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits 0 is jointly continuous in \(x\) and \(\delta\), provided \(\delta \leq 0\). As an application, we show that the \(\operatorname{SLE}(\kappa)\) welding homeomorphism is continuous in \(\kappa\) for \(\kappa \in [0,4]\). Our motivation behind this is to study the well known problem of the continuity of \(\operatorname{SLE} \kappa\) in \(\kappa\). The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.Identification of the polaron measure in strong coupling and the Pekar variational formula.https://www.zbmath.org/1456.602112021-04-16T16:22:00+00:00"Mukherjee, Chiranjib"https://www.zbmath.org/authors/?q=ai:mukherjee.chiranjib"Varadhan, S. R. S."https://www.zbmath.org/authors/?q=ai:varadhan.s-r-srinivasaSummary: The path measure corresponding to the Fröhlich polaron appearing in quantum statistical mechanics is defined as the tilted measure
\[
\text{d} \widehat{\mathbb{P}}_{\varepsilon, T} = \frac{1}{Z (\varepsilon, T)} \exp\left\{\frac{1}{2} \int\nolimits_{-T}^T \int_{-T}^T \frac{\varepsilon \text{e}^{-\varepsilon |t - s|}}{|\omega (t) - \omega (s)|} \text{d}s \text{d}t\right\} \text{d}\mathbb{P}.
\]
Here, \(\varepsilon > 0\) is a constant known as the Kac parameter or the inverse-coupling parameter, and \(\mathbb{P}\) is the distribution of the increments of the three-dimensional Brownian motion. In [the authors, Commun. Pure Appl. Math. 73, No. 2, 350--383 (2020; Zbl 1442.60082)] it was shown that, when \(\varepsilon > 0\) is sufficiently small or sufficiently large, the (thermodynamic) limit \(\lim_{T \to \infty} \widehat{\mathbb{P}}_{\varepsilon, T} = \widehat{\mathbb{P}}_{\varepsilon}\) exists as a process with stationary increments, and this limit was identified explicitly as a mixture of Gaussian processes. In the present article the \textit{strong coupling limit} or the vanishing Kac parameter limit \(\lim_{\varepsilon \to 0} \widehat{\mathbb{P}}_{\varepsilon}\) is investigated. It is shown that this limit exists and coincides with the increments of the so-called \textit{Pekar process}, a stationary diffusion with generator \(\frac{1}{2} \Delta +(\nabla \psi / \psi) \cdot \nabla\), where \(\psi\) is the unique (up to spatial translations) maximizer of the Pekar variational problem
\[
g_0 = \underset{\| \psi \|_2 = 1}{\text{sup}} \left\{\int\nolimits_{\mathbb{R}^3} \int\nolimits_{\mathbb{R}^3} \psi^2(x) \psi^2(y) |x - y|^{-1} \text{d}x \text{d}y - \frac{1}{2} \|\nabla \psi\|_2^2\right\}.
\]
As the Pekar process was also earlier shown [the authors, Ann. Probab. 44, No. 6, 3934--3964 (2016; Zbl 1364.60037); \textit{W. König} and the first author, Ann. Inst. Henri Poincaré, Probab. Stat. 53, No. 4, 2214--2228 (2017; Zbl 1382.60107)] to be the limiting object of the mean-field polaron measures, the present identification of the strong coupling limit is a rigorous justification of the mean-field approximation of the polaron problem (on the level of path measures) conjectured by \textit{H. Spohn} in [``Effective mass of the polaron: A functional integral approach'', Ann. Physics 175, 278--318 (1987)]. Replacing the Coulomb potential by continuous function vanishing at infinity and assuming uniqueness (modulo translations) of the relevant variational problem, our proof also shows that path measures coming from a Kac interaction of the above form with translation invariance in space converge to the increments of the corresponding mean-field model.Directed, cylindric and radial Brownian webs.https://www.zbmath.org/1456.602062021-04-16T16:22:00+00:00"Coupier, David"https://www.zbmath.org/authors/?q=ai:coupier.david"Marckert, Jean-François"https://www.zbmath.org/authors/?q=ai:marckert.jean-francois"Tran, Viet Chi"https://www.zbmath.org/authors/?q=ai:tran.viet-chiSummary: The Brownian web (BW) is a collection of coalescing Brownian paths \((W_{(x,t)},(x,t) \in \mathbb{R} ^2)\) indexed by the plane. It appears in particular as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in \textit{C.F. Coletti} and \textit{L.A. Valencia} [``Scaling limit for a family of random paths with radial behavior'', Preprint, \url{arXiv:1310.6929}] is shown to converge to the CBW.Discovering transition phenomena from data of stochastic dynamical systems with Lévy noise.https://www.zbmath.org/1456.370862021-04-16T16:22:00+00:00"Lu, Yubin"https://www.zbmath.org/authors/?q=ai:lu.yubin"Duan, Jinqiao"https://www.zbmath.org/authors/?q=ai:duan.jinqiaoSummary: It is a challenging issue to analyze complex dynamics from observed and simulated data. An advantage of extracting dynamic behaviors from data is that this approach enables the investigation of nonlinear phenomena whose mathematical models are unavailable. The purpose of this present work is to extract information about transition phenomena (e.g., mean exit time and escape probability) from data of stochastic differential equations with non-Gaussian Lévy noise. As a tool in describing dynamical systems, the Koopman semigroup transforms a nonlinear system into a linear system, but at the cost of elevating a finite dimensional problem into an infinite dimensional one. In spite of this, using the relation between the stochastic Koopman semigroup and the infinitesimal generator of a stochastic differential equation, we learn the mean exit time and escape probability from data. Specifically, we first obtain a finite dimensional approximation of the infinitesimal generator by an extended dynamic mode decomposition algorithm. Then, we identify the drift coefficient, diffusion coefficient, and anomalous diffusion coefficient for the stochastic differential equation. Finally, we compute the mean exit time and escape probability by finite difference discretization of the associated nonlocal partial differential equations. This approach is applicable in extracting transition information from data of stochastic differential equations with either (Gaussian) Brownian motion or (non-Gaussian) Lévy motion. We present one- and two-dimensional examples to demonstrate the effectiveness of our approach.
{\copyright 2020 American Institute of Physics}Algebraic and arithmetic area for \(m\) planar Brownian paths.https://www.zbmath.org/1456.602072021-04-16T16:22:00+00:00"Desbois, Jean"https://www.zbmath.org/authors/?q=ai:desbois.jean"Ouvry, Stéphane"https://www.zbmath.org/authors/?q=ai:ouvry.stephaneOptimal strategy to capture a skittish Lamb wandering near a precipice.https://www.zbmath.org/1456.826532021-04-16T16:22:00+00:00"Chupeau, M."https://www.zbmath.org/authors/?q=ai:chupeau.marie"Bénichou, O."https://www.zbmath.org/authors/?q=ai:benichou.olivier"Redner, S."https://www.zbmath.org/authors/?q=ai:redner.sidneyFractal transformed doubly reflected Brownian motions.https://www.zbmath.org/1456.602092021-04-16T16:22:00+00:00"Ehnes, Tim"https://www.zbmath.org/authors/?q=ai:ehnes.tim"Freiberg, Uta"https://www.zbmath.org/authors/?q=ai:freiberg.uta-renataTwo-time correlation and occupation time for the Brownian bridge and tied-down renewal processes.https://www.zbmath.org/1456.602102021-04-16T16:22:00+00:00"Godrèche, Claude"https://www.zbmath.org/authors/?q=ai:godreche.claudeA Lagrangian scheme for numerical evaluation of the noncausal stochastic integral.https://www.zbmath.org/1456.601352021-04-16T16:22:00+00:00"Ogawa, Shigeyoshi"https://www.zbmath.org/authors/?q=ai:ogawa.shigeyoshiSummary: We are concerned with a noncausal approach to the numerical evaluation of the stochastic integral \(\int f dW_t\) with respect to Brownian motion. Viewed as a special case of the numerical solution (in strong sense) of the SDE, it may be believed that the precision level of such an approximation scheme that uses only a finite number of increments \(\Delta_kW=W(t_{k+1})-W(t_k)\) of Brownian motion, would not exceed the order \(O\big (\frac{1}{n}\big )\) where \(n\) is the number of steps for discretization. We present in this note a simple but not trivial example showing that this belief is not correct. The discussion is developed on the basis of the noncausal theory of stochastic calculus introduced by the author.