Friday, November 20th at 12:30pm, Virtual Room (Zoom Meeting ID: 754 716 1091)

"Branching Random Walks on Multidimensional Lattices"
Dr. Elena Yarovaya, Lomonosov Moscow State University, Russia

Abstract:

The talk is devoted to continuous-time stochastic processes, which may be describe in terms of birth, death, and walking of particles on multidimensional lattices. Such processes are called branching random walks (BRWs), and the points of the lattice, at which the birth and death of particles can occur, are called sources of branching. The talk provides a series of results on the asymptotic behavior of the particle numbers and their moments for symmetric BRWs with one source of branching and a finite or infinite number of the initial particles under different assumptions on the variance of random walk jumps. The proof of some limit theorems on BRWs with a finite number of sources and pseudo-sources, admitting possible violation of symmetry of an underlying random walk, is based on applying of Carleman’s condition, which is guaranteed the uniqueness of determining the limiting probability distribution of the number of particles at the lattice points by its moments. The problems of a relationship between such sufficient conditions based on the growth rate of the limiting moments of the number of particles at the lattice points are discussed. For BRWs with sources of branching at each lattice point, in which the reproduction law of particles is described by a critical branching process, theorems on the behavior of populations and subpopulations of particles are obtained. A series of simulation results of BRWs will be presented.