Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

TL;DR

This paper introduces a new model of a multidimensional branching random walk with a single reproduction center and multiple absorption points, analyzing its long-term growth behavior based on spectral properties.

Contribution

The paper develops a novel branching random walk model with absorption and characterizes its asymptotic behavior using spectral analysis of the evolution operator.

Findings

Exponential growth of particle numbers under certain spectral conditions

Asymptotic behavior depends on the eigenvalues of the evolution operator

Limit theorems established for the total and local particle counts

Abstract

We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only absorption of particles can occur. The asymptotic behavior of the integer moments of both the total number of particles and the number of particles at a lattice point is studied depending on the relationship between the model parameters. In the case of the existence of an isolated positive eigenvalue of the evolution operator of the average number of particles, a limit theorem is obtained on the exponential growth of both the total number of particles and the number of particles at a lattice point.