Page 184 - Special Topic Session (STS) - Volume 4

STS579 Anastasiia Rytova et al.
survival probability may be avoiding this area due to the complication of the
return conditions for a particle, such as an increase the dimension of space or
an increase in the length of the particle displacement. An example of the
relevant model and conditions will be given in sections 3 and 4.
There are many combinations of assumptions on branching random walks
(BRW) models: discreteness or continuity of time, discreteness, compactness
and dimension of space, number and configuration of branching sources,
number and location of initial particles, random walk and branching process
properties. In this paper we consider a symmetric continuous time BRWs on
the lattices ℤ , ≥ 1 with a finite number of branching sources and a single

initial particle. Suppose that particle performs a random walk on points of ℤ

until reaching one of the branching sources, where it can die or give a random
number of descendants, and then each one evolves according to the same
rules independently of each other. The underlying random walk is assumed to
be symmetric, spatially homogeneous and irreducible. As a rule, such models
were considered under the assumption, that the variance of random walk
jumps is finite, therefore the new effects of infinite variance condition are
expected. One of the first works that investigated a model for a simple random
walk with one source and pure birth was, apparently, the work Yarovaya (1991).
More general cases of symmetric BRWs with finite variance of jumps and one
branching source have been studied by many authors (e.g., Albeverio,
Bogachev, & Yarovaya (1998)). As was shown in Yarovaya (2013a), rejection of
jump variance finiteness assumption leads to new effects for BRW. A number
of publications were devoted to random walks with an infinite variance of
jumps (see Borovkov & Borovkov (2008) and detailed bibliography therein). In
Agbor, Molchanov & Vainberg (2015), for random walks on ℤ , ≥ 1, with

heavy tails and appropriate regularity conditions, a global limit theorem on
the behavior of the transition probabilities with simultaneous growth of time
and lattice coordinates was obtained.

2. Methodology
Let us give a brief description of the model proposed in Yarovaya (2013a).
The underlying random walk is specified by a matrix = ((, )) of
,∈ℤ
transition intensities satisfying the conditions (, ) ≥ 0 if ≠ , −∞ <
(, ) < 0, ∑ ∈ℤ (, ) = 0 symmetry and spatially homogeneity (, ) =
(, ) = (0, − ). Then the values of (, ) can be expressed by the
function of one argument () ≔ (0, ). Assume irreducibility of the random
walk, which means that for every ∈ ℤ , there exists a set of vectors

, … , ∈ ℤ such that = ∑ and ( ) ≠ 0 for 1 ≤ ≤ . We denote by

1
=1

the variance of jumps of a random walk, then
2
()
2
≔ ∑|| 2 . (1)
−(0)
≠0
173 | I S I W S C 2 0 1 9