Page 176 - Special Topic Session (STS) - Volume 4

STS579 Daria Balashova
It is convenient to represent the evolution equations for the transition
probabilities and the numbers of particles in the form of linear differential
equations in Banach spaces. By virtue of their linearity, the study of the
asymptotics of solutions as → ∞ leads to the study of the spectrum of the
corresponding operators. In particular, the presence of a positive eigenvalue
in the spectrum of an evolutionary operator ensures an exponential growth in
the number of particles both at each point and in the entire lattice [10]. BRW
with an exponential increase in the number of particles are called supercritical.
Analysis of the evolutionary operator of BRW with several sources in
general was carried out in [10], where in particular it was noted that the
presence of branching sources can lead to the appearance of positive operator
eigenvalues. In [2] it was proved that for the case of equal source intensities
and finite variance of jumps, the number of eigenvalues (including multiplicity)
does not exceed the number of sources and the multiplicity of each
eigenvalue does not exceed − 1.
In this paper, we consider BRW in which the birth and death of particles
can be specified by a subcritical. The sources can be of different intensities,
positive where birth prevails over death, and negative where the opposite is
true. Analyzing the corresponding equations is analytically quite difficult.
Therefore, steps have been taken in modelling such processes, which make it
possible to estimate numerically the values for critical boundaries for source
intensities.

2. Methodology
We consider BRW on the multidimensional lattice ℤ , ≥ 1, in which

branching — birth or death — occurs in the sources , , … , . We assume

2
1
that random walk is given by a matrix of transient intensities

with the properties (, ) = (, ) = (0, − ) = ( − ) for all and .
Thus, a random walk is symmetric and spatially homogeneous. Moreover, we
assume the regularity properties ∑ ∈ () and irreducibility are fulfilled, i.e.

for all ∈ ℤ there exists a set of vectors , , … , ∈ ℤ such that =∑
2

1
=1
and ( ) ≠ 0 for = 1,2, … , .

The transition probability (,∙, ) is conveniently considered as a function
() in (ℤ ), depending on the time and parameter . For the time ℎ → 0
2

the following equalities hold:

(1)

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