Martin boundaries and asymptotic behavior of branching random walks

TL;DR

This paper explores the asymptotic behavior of supercritical branching random walks on infinite graphs, revealing new links between Martin boundaries and survival/extinction phenomena in subgraphs.

Contribution

It establishes novel connections between $t$-Martin and standard Martin boundaries, and analyzes survival and persistence of branching random walks in subgraphs, with various examples.

Findings

Branching random walk trajectories have typical asymptotic directions.

New relationship between $t$-Martin and standard Martin boundaries.

Survival in subgraphs can occur even if the underlying random walk does not stay in the subgraph.

Abstract

Let $G$ be an infinite, locally finite graph. We investigate the relation between supercritical, transient branching random walk and the Martin boundary of its underlying random walk. We show results regarding the typical asymptotic directions taken by the particles, and as a consequence we find a new connection between $t$-Martin boundaries and standard Martin boundaries. Moreover, given a subgraph $U$ we study two aspects of branching random walks on $U$: when the trajectories visit $U$ infinitely often (survival) and when they stay inside $U$ forever (persistence). We show that there are cases, when $U$ is not connected, where the branching random walk does not survive in $U$, but the random walk on $G$ converges to the boundary of $U$ with positive probability. In contrast, the branching random walk can survive in $U$ even though the random walk eventually exits $U$ almost surely. We provide several examples and counterexamples.