Subcritical catalytic branching random walk with finite or infinite variance of offspring number

TL;DR

This paper investigates the asymptotic behavior of local particle distributions in subcritical catalytic branching random walks on lattices, extending previous work to cases with offspring distributions of finite or infinite variance.

Contribution

It introduces new theorems for subcritical regimes with offspring moments of order 1+δ, using fractional moments and derivatives, expanding understanding beyond supercritical and critical cases.

Findings

Established asymptotic distribution behaviors of local particles

Connected fractional moments with fractional derivatives of Laplace transforms

Extended analysis to offspring distributions with infinite variance

Abstract

Subcritical catalytic branching random walk on d-dimensional lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particles numbers are established. To prove the results different approaches are used including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated assuming finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the present paper for the offspring number in subcritical regime finiteness of the moment of order 1+\delta is required where \delta is some positive number. Keywords and phrases: branching random walk, subcritical regime, finite variance of offspring number, infinite variance of offspring number, local particles numbers, limit theorems, fractional moments, fractional derivatives.