Page 186 - Special Topic Session (STS) - Volume 4

STS579 Anastasiia Rytova et al.
with initial condition (0, , ) = ( − ), (0, , ) = ( − ), (, ) =

1 respectively, where for ∈ (ℤ ),1 ≤ ≤ ∞ , the operator : (ℤ ) →
(ℤ ) is as follows ()() = ∑ (, )( ) , and the operator is

′
′
′

specified by the equality = + ∑ ∆ , where ∆ = , and = (∙)
=1

denotes the column-vector on the lattice taking a unit value at the point and
vanishing elsewhere. In (ℤ )the operator is self-adjoint.

2
To investigate the population survival probability, we denote (, ) ≔
{ > 0} as the probability of the presence of at least one particle on the

lattice. By work Yarovaya (2009), we have the equation for BRW with the single
branching source located at the 0 ∈ ℤ , that is valid for BRW regardless of the

finiteness variance of jumps

(, ) = 1 − ∫ ( − , , 0)(1 − (, 0)), (7)
0
where (0, ) = 1.
It is convenient to study the BRW in terms of the so-called Green’s function
∞ −
(, ) ≔ ∫ (, , ) , which is the Laplace transform of the random

0
walk transition probability. Following, for example, Yarovaya (2013a), a
random walk will be called recurrent in the case when ≔ (0,0) = ∞ and
0
0
transient in the case when < ∞. The behavior of a BRW essentially depends
0
on the recurrence property.
We denote by and called critical (for BRW) intensity of the branching

source the lowest intensity of the source such that for > the spectrum
of the operator contains a positive eigenvalue. In work Albeverio,

Bogachev, & Yarovaya (1998), for the BRW with a finite variance of jumps, it
was established that the particle numbers, both at each lattice point and on
the entire lattice, grow exponentially only for > . In this sense, the BRW
with > can be called supercritical, with = critical and with <
subcritical BRW. In Yarovaya (2015), it was shown that for BRW with
branching sources, regardless of the assumptions on the variance of jumps,
the following relations hold: if = ∞ then = 0 for > 1, and if = ∞
0
0
then = 0 −1 for = 1 and 0 < < 0 −1 for ≥ 2.
For BRW model with a single branching source, the rejection of the
variance of jumps finiteness leads to new effects. The properties and
asymptotics of the function (, , ) and, as a consequence, the functions
(, ) and (, , ) change qualitatively. In particular, the asymptotics as

→ ∞ of transition probability are
⁄
− 2 ,
(, , )~ {
⁄
ℎ , − , ∈ (0,2) ℎ − ,

where , ℎ , > 0 are from Yarovaya (2007), Rytova & Yarovaya (2016)

respectively. Therefore for heavy- tailed BRW the relation < ∞ is possible
0
in the dimension = 1 for ∈ (0, 1), and also in dimensions ≥ 2 for ∈
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