A criterion for transience of multidimensional branching random walk in random environment

TL;DR

This paper establishes an explicit criterion for determining the transience of multidimensional branching random walks in random environments, confirming a conjecture and providing insights into critical cases.

Contribution

It introduces a precise transience criterion for multi-dimensional BRWRE, confirming the necessity and sufficiency of Condition L and analyzing critical cases.

Findings

Condition L is necessary and sufficient for transience.

Critical branching random walk is either transient or dies out locally.

The criterion applies to key classes of branching random walks.

Abstract

We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that \emph{Condition L} of Comets and Popov is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branching random walks and implies that the critical branching random walk is transient resp. dies out locally.