Page 175 - Special Topic Session (STS) - Volume 4

STS579 Daria Balashova

Simulation of branching random walks with
different intensity of branching sources
Daria Balashova
Lomonosov Moscow State University, Moscow, Russia

Abstract
It is a common practice to describe branching random walks in terms of birth,
death and walk of particles, which makes it easier to use them in different
applications. We consider a continuous-time symmetric supercritical
branching random walk on a multidimensional lattice with a finite set of
particle generation centers, i.e. branching sources. It is useful in different
applications such as statistical physics, population dynamics and chemical
kinetics. The branching of particles occurs at lattice points called branching
sources and is determined by the continuous-time Galton-Watson process.
The intensity of the source as the first derivative of infinitesimal generating
function of the Galton-Watson process is the quantitative characteristic of the
average number of particle descendants that are born in it. Existence of a
positive eigenvalue of the evolutionary operator of the average number of
particles involves the exponential growth of the first moment of the total
number of particles both at an arbitrary point and on the entire lattice. Main
attention is paid to the case when sources with positive and negative
intensities are in an arbitrary configuration and, as an example, in a simplex.
For applied research, behavior at finite time intervals is required. In addition
to the limit theorems, the approach based on simulations by the Monte Carlo
method is considered in the talk.

Keywords
branching processes; random walks; multidimensional lattices; simulation.

1. Introduction
A branching random walk (BRW) with continuous time on a
multidimensional lattice ℤ , ≥ 1, with a finite number of branching sources

on it is considered. We assume that the walk is homogeneous in time and
space, symmetrical and irreducible. Branching processes are used to describe
the population dynamics of objects with non-overlapping generations, for
example, to describe the spread of viral infections, see [1], [4], modeling the
development of the epidemic and the effects of vaccination, [3] and [7], as well
as biological and genetic systems [8]. The introduction of a walk will allow to
describe the spatial distribution of such processes.

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