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Title: Fundamental problems arising in the analysis of applied stochastic models Authors:  Elena Yarovaya - Lomonosov Moscow State University (Russia) [presenting]
Abstract: The aim is to study the behavior of some stochastic processes describing the evolution of systems with a complex structure. The main focus is on models involving the generation and transport of particles, so-called branching random walks. In recent years, branching random walks have become a rapidly developing area of stochastic processes. Special attention is paid to the analysis of the asymptotic behavior of particle numbers and their moments for symmetric BRWs with a few sources of branching and a finite or infinite number of initial particles under various assumptions on the variance of random walk jumps. The proofs of some limit theorems for BRWs with a finite number of sources and pseudo-sources, where violation of random walk symmetry is allowed, are often based on the verification of the Carleman condition, which guarantees the uniqueness of the definition of the limit probability distribution of particle numbers by their moments. In this context, questions about the relationship between such sufficient conditions based on the growth rate of the limiting moments of the particle numbers are discussed. For BRWs with branching sources at each point of the lattice, where the law of reproduction and death of particles is described by a critical branching process, limit theorems for the behavior of populations and subpopulations of particles are given. The research was supported by RFBR, project No. 20-01-00487.