Asymptotic Behavior of Local Particles Numbers in Branching Random Walk

TL;DR

This paper analyzes the long-term behavior of local particle counts in a critical catalytic branching random walk on integer lattices, providing asymptotic results, probability estimates, and a Yaglom limit theorem.

Contribution

It introduces a comprehensive asymptotic analysis for local particles in critical catalytic branching random walks, including new limit theorems and probabilistic estimates.

Findings

Asymptotic behavior of mean local particles numbers determined

Probability of particles at a fixed point characterized asymptotically

Yaglom type limit theorem established for local particle counts

Abstract

Critical catalytic branching random walk on d-dimensional integer lattice is investigated for all d. The branching may occur at the origin only and the start point is arbitrary. The asymptotic behavior, as time grows to infinity, is determined for the mean local particles numbers. The same problem is solved for the probability of particles presence at a fixed lattice point. Moreover, the Yaglom type limit theorem is established for the local number of particles. Our analysis involves construction of an auxiliary Bellman-Harris branching process with six types of particles. The proofs employ the asymptotic properties of the (improper) c.d.f. of hitting times with taboo. The latter notion was recently introduced by the author for a non-branching random walk on an integer lattice.