Page 168 - Special Topic Session (STS) - Volume 4

STS579 Elena Yarovaya
One of the principal problems in BRW models is a study of the evolution
of the field of particles on the entire lattice. The methods of the spectral theory
of operators with multipoint perturbations, see (Yarovaya, 2017a), will be
applied to the study of its evolution. It will be developed the results for weakly
supercritical BRWs on ℤ , ≥ 1, obtained in (Yarovaya, 2017c). The methods

of the theory of large deviations for BRWs, see (Molchanov and Yarovaya,
2013; Agbor et al., 2015), will be used to study the distribution of population
inside the front of propagation of the weekly supercritical BRW. In the frame
of the proposed models, it will be undertaken the spatio-temporal analysis of
the system. These results help to analyze the distribution of the population
inside the propagation front of particles for a weekly supercritical BRW.

2. Methodology
A BRW is a stochastic process combining in itself random walk of particles
on ℤ with their branching at some lattice points of ℤ , ≥ 1. A random

walk of particles on ℤ is defined in terms of the matrix of transition

intensities = (, )) ,∈ℤ , which features the regularity property:
∑ (, ) = 0 for all , where (, ) ≥ 0 for ≠ and (, ) < 0. We
∈ℤ
assume that the intensities (, ) are symmetric and spatially homogeneous;
that is, ( − ) ≔ (, ) = (, ) = (0, − ) and a random walk is
irreducible: for each ∈ ℤ there exists a set of vectors … , ∈ ℤ such that

,
1,
= ∑ and . ( ) ≠ 0 for = 1, … , . Birth and death of particles may

=1
occur at some points of the lattice … , . The branching mechanism at each
1,

source = 1, … , , is controlled by a Galton Watson continuous-time process,
,
which is defined by the infinitesimal generating function (, ) =

∑ ∞ ( ) , 0 ≤ ≤ 1, where ( ) ≥ 0 for ≠ 1, ( ) < 0 and

=0
1

∑ ( ) = 0. It is assumed that each of the particles evolves independently of

the rest of particles. We note that the condition for finiteness of all moments,
that is, () (1, ) < ∞ for all ∈ ℕ is essentially used in some proofs of the limit

theorems on behavior of the numbers of particles in BRW (see, for example,
(Yarovaya, 2007)). Put ≔ (1, ) for every .
(′)

In the BRW models (Yarovaya, 2012), multipoint perturbations of the
generator of symmetrical random walk .0' arise which in the case of identical
intensity of the sources are given by

(1)

where ∈ ℤ , : (ℤ ) → (ℤ ), ∈ [1, ∞], is a symmetrical operator

generated by the matrix and obeying the formula

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