Recent zbMATH articles in MSC 37A50 https://www.zbmath.org/atom/cc/37A50 2022-01-14T13:23:02.489162Z Werkzeug Random evolution equations: well-posedness, asymptotics, and applications to graphs https://www.zbmath.org/1475.35428 2022-01-14T13:23:02.489162Z "Bonaccorsi, Stefano" https://www.zbmath.org/authors/?q=ai:bonaccorsi.stefano "Cottini, Francesca" https://www.zbmath.org/authors/?q=ai:cottini.francesca "Mugnolo, Delio" https://www.zbmath.org/authors/?q=ai:mugnolo.delio Summary: We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for pathwise convergence in norm of the (random) propagator towards a (deterministic) steady state. We apply our findings in two environments with randomly evolving features: ensembles of difference operators on combinatorial graphs, or else of differential operators on metric graphs. Large gap asymptotics for Airy kernel determinants with discontinuities https://www.zbmath.org/1475.37014 2022-01-14T13:23:02.489162Z "Charlier, Christophe" https://www.zbmath.org/authors/?q=ai:charlier.christophe "Claeys, Tom" https://www.zbmath.org/authors/?q=ai:claeys.tom The so-called Airy point process is a very important object that represents a collection of random points on the real line. This process arises as a scaling limit in different probabilistic models, and is said to be universal. It is an instance of a determinantal point process for which the kernel is specified by an Airy function. One can also study this process after removing some of its points. This gives rise to the so-called thinned Airy process, which is exactly the model considered in this paper. If $$\overline{x}$$ are some points $$x_{1},\ldots, x_{m}$$ on the real line, and $$\overline{s}$$ are values $$s_{1},\ldots, s_{m}$$ in $$(0,1)$$, the function $$F(\overline{x},\overline{s})$$ represents the probability having a gap on the interval $$(x_{m},\infty)$$ in the thinned Airy process, in which the particle in $$(x_{j}, x_{j-1})$$ is removed using the probability $$s_{j}$$. The authors provide some asymptotic results for the function $$F(\overline{x},\overline{s})$$ when the vector $$\overline{x}$$ becomes very large. The main technique used for the analysis is based on the Hilbert-Riemann problem device, a well-known tool in complex analysis. The authors also compare their results with formulas previously found in the literature, in particular when $$m=1$$. Reviewer: Carlos Gabriel Pacheco (Ciudad de México) Targets and holes https://www.zbmath.org/1475.37015 2022-01-14T13:23:02.489162Z "Giulietti, P." https://www.zbmath.org/authors/?q=ai:giulietti.paolo "Koltai, P." https://www.zbmath.org/authors/?q=ai:koltai.peter "Vaienti, S." https://www.zbmath.org/authors/?q=ai:vaienti.sandro The authors consider a dynamical system with an absorbing region, a hole $$H$$, such that an orbit entering $$H$$ terminates its evolution. By considering the orbits of the whole state space, a surviving set is constructed within which a point and a small ball around it, the target set $$B$$, are fixed. The probability of hitting $$B$$ for the first time after $$n$$ steps while avoiding $$H$$, is investigated, as $$n \rightarrow \infty$$. The entrance of the system trajectory into the target is called an extreme event, and the closest approach of the trajectory to the target is measured by extreme values of a suitable function of the distance. An extreme value distribution is obtained using a spectral approach on suitably perturbed transfer operators. The boundary levels and the extremal index of the extreme value distribution are expressed in terms of the Hausdorff distance of the surviving set and of the escape rate, respectively. The extreme value distribution explicitly depends on whether the target point in the surviving set is periodic or not. The extremal index is computed explicitly and it is shown how a degenerate extreme value distribution arises when the target set becomes disjoint from the surviving sets. Reviewer: Ravi Sreenivasan (Mysore) Stochastic heat equations for infinite strings with values in a manifold https://www.zbmath.org/1475.37088 2022-01-14T13:23:02.489162Z "Chen, Xin" https://www.zbmath.org/authors/?q=ai:chen.xin.1 "Wu, Bo" https://www.zbmath.org/authors/?q=ai:wu.bo "Zhu, Rongchan" https://www.zbmath.org/authors/?q=ai:zhu.rongchan "Zhu, Xiangchan" https://www.zbmath.org/authors/?q=ai:zhu.xiang-chan The authors construct a stochastic process representing the evolution of a one- or two-sided infinite string, forced by space-time white noise, in a complete and stochastically complete manifold. Formally, this is a stochastic heat equation corresponding to the Dirichlet energy defined using the metric on the manifold. The paper continues the program begun in [\textit{M. Röckner} et al., SIAM J. Math. Anal. 52, No. 3, 2237--2274 (2020; Zbl 1445.60048)], which considers the case of a compact string. The infinite volume of the domain necessitates constructing the process in weighted $$L^{2}$$ spaces on the half-line or the full line. In addition to the construction, the authors establish functional inequalities providing a partial characterization of the ergodic properties of the string process in terms of the curvature of the manifold. Reviewer: Alexander Dunlap (New York) Anticipation decides on lane formation in pedestrian counterflow -- a simulation study https://www.zbmath.org/1475.37107 2022-01-14T13:23:02.489162Z "Cirillo, Emilio N. M." https://www.zbmath.org/authors/?q=ai:cirillo.emilio-nicola-maria "Muntean, Adrian" https://www.zbmath.org/authors/?q=ai:muntean.adrian Summary: Human crowds base most of their behavioral decisions upon anticipated states of their walking environment. We explore a minimal version of a lattice model to study lanes formation in pedestrian counterflow. Using the concept of horizon depth, our simulation results suggest that the anticipation effect together with the presence of a small background noise play an important role in promoting collective behaviors in a counterflow setup. These ingredients facilitate the formation of seemingly stable lanes and ensure the ergodicity of the system. Averaging principle for impulsive stochastic partial differential equations https://www.zbmath.org/1475.60122 2022-01-14T13:23:02.489162Z "Liu, Jiankang" https://www.zbmath.org/authors/?q=ai:liu.jiankang "Xu, Wei" https://www.zbmath.org/authors/?q=ai:xu.wei.1 "Guo, Qin" https://www.zbmath.org/authors/?q=ai:guo.qin On sub-geometric ergodicity of diffusion processes https://www.zbmath.org/1475.60149 2022-01-14T13:23:02.489162Z "Lazić, Petra" https://www.zbmath.org/authors/?q=ai:lazic.petra "Sandrić, Nikola" https://www.zbmath.org/authors/?q=ai:sandric.nikola Summary: In this article, we discuss ergodicity properties of a diffusion process given through an Itô stochastic differential equation. We identify conditions on the drift and diffusion coefficients which result in sub-geometric ergodicity of the corresponding semigroup with respect to the total variation distance. We also prove sub-geometric contractivity and ergodicity of the semigroup under a class of Wasserstein distances. Finally, we discuss sub-geometric ergodicity of two classes of Markov processes with jumps.