Strong local survival of branching random walks is not monotone

TL;DR

This paper investigates the properties of strong local survival in branching random walks, revealing non-monotonic behavior in continuous time and providing counterexamples that challenge previous assumptions about survival phases.

Contribution

It demonstrates that strong local survival is not necessarily monotone in continuous-time branching random walks and provides counterexamples to existing beliefs about survival phases.

Findings

Strong local survival is non-monotone in continuous time.

Existence of an irreducible BRW with site-dependent strong local survival.

The generating function can have more than two fixed points, contradicting prior results.

Abstract

The aim of this paper is the study of the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results about the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasi transitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit non-strong local survival. Finally we show that the generating function of a irreducible BRW can have more than two fixed points; this disproves a previously known result.