Page 167 - Special Topic Session (STS) - Volume 4

STS579 Elena Yarovaya

Stochastic evolutionary system on
multidimensional lattices
Elena Yarovaya
Lomonosov Moscow State University, Moscow, Russia

Abstract
For the study of stochastic evolution of particle systems on a non-compact
phase space we apply an approach focused on continuous-time branching
random walks on multidimensional lattices. The main object of study is the
limit distribution of particles on the lattice. Special attention is paid to
branching random walks with large deviations. The limit theorems on
asymptotic behavior of the Green function for transition probabilities were
established for random walks with both a finite and infinite variance of jumps.
The obtained results allow to study the front of branching random walk and
the structure of the particle population inside of the front and near to its
boundary. For supercritical branching random walks, it is shown that the
amount of positive eigenvalues of the evolutionary operator, counting their
multiplicity, does not exceed the amount of branching sources on the lattice,
while the maximal of these eigenvalues is always simple. 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. The presented results are based on
Green’s function representation of transition probabilities of an underlying
random walk and cover not only the case of the finite variance of jumps but
also a less studied case of infinite variance of jumps.

Keywords
branching random walks; limit distributions of particles; the spectrum of the
evolutionary operator; Green’s function; large deviations.

1. Introduction
We offer to use models of branching random walks (BRWs) for a study of
the dynamics of stochastic lattice systems. Continuous-time BRWs on
multidimensional lattices provide an important example of stochastic
multicompartment systems in which the evolutionary processes depend on
the spatial dynamics and the structure of a medium. The dynamics of such
processes is usually described in terms of birth, death and walks of particles
on the lattice ℤ , ≥ 1. The structure of a medium is defined by the particle

offspring reproduction law at the lattice points called branching sources. Such
a description covers various applications of BRWs (Zel’dovich et al., 1988;
Cranston et al., 2009; Ermakova et al., 2019).

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