B1517
Title: Stochastic evolution of particle systems for branching random walks in non homogeneous and homogeneous environments
Authors: Elena Yarovaya - Lomonosov Moscow State University (Russia) [presenting]
Abstract: For the study of stochastic evolution of particle systems, an approach is applied which is focused on continuous-time symmetric branching random walks on multidimensional lattices. The main object of study is the limit distribution of particles on the lattice and their moments. The limit theorems on the asymptotic behavior of the Green function for transition probabilities were established for random walks under different assumptions on a variance of random walk jumps. For supercritical branching random walks with one initial particle and a finite number of particle generation centers called branching sources and located at the lattice points, it is shown that the amount of positive eigenvalues of the evolutionary operator of the mean number of particles, counting their multiplicity, does not exceed the amount of branching sources with a positive intensity on the lattice. We demonstrate that the appearance of multiple lower eigenvalues in the spectrum of the evolutionary operator can be caused by a kind of `symmetry' in the spatial configuration of branching sources. If initially there is one particle at each lattice point which can walk over the lattice and produce offspring at every lattice point by a critical Markov branching process under the assumption that the underlying walk is recurrent, the convergence of the distribution of the particle field to the limit stationary distribution is obtained.
