Strong Local Survival of Branching Random Walks is Not Monotone

Abstract

In this paper we study 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 for 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 quasitransitive). 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 nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

The probabilities of extinction in a branching random walk on a strip

“…Throughout this paper we make repeated use of [1,Theorem 3.3] which, for completeness, we now state and prove. Theorem 1.1.…”

“…It is well known that q is the minimal nonnegative solution of the fixed-point equation (1) for a formal definition). When the set X is finite, many of the fundamental questions concerning q are resolved in classical texts such as [10].…”

“…The vectors q(A) are therefore interesting in their own right. Apart from the recent work in [1,2] which did not directly address the possibility that q < q(A) <q, it appears that the vectors q(A) have received little attention in the literature. In this more general context, Assertions (i)-(iii) above lead to a number of natural questions: (i) can we use M to determine whether q < q(A) <q?…”

Extinction probabilities in branching processes with countably many types: a general framework

“…Question (ii). Recent work addresses related questions: in Bertacchi and Zucca (2014) and Bertacchi and Zucca (2020), the authors provide equivalent conditions for qpAq " q for every non-empty A Ď X ; in Braunsteins and Hautphenne (2020), the authors give sufficient conditions for qpAq ď qpBq that apply to any MGWBP and A, B Ď X , as well as sufficient conditions for q ă qpAq ă q that apply to block LHBPs. In Theorem 4.1 we present a number of necessary and sufficient conditions for q x pAq ă q x pBq for any initial type x; this is a significant improvement on Bertacchi and Zucca (2014, Theorem 3.3) and Bertacchi and Zucca (2020, Theorem 2.4) (see Section 4 for details).…”

“…When q ă q, the set of extinction probability vectors Ext may contain more than two distinct elements. For instance, in processes that exhibit non-strong local survival (q ă q ă 1 " qpHq), Ext contains at least three distinct elements; see for instance Bertacchi and Zucca (2014); Gantert et al (2010); Menshikov and Volkov (1997); Müller (2008) for examples of such processes. In recent years, various examples with more than three extinction probabilities have been constructed: for instance, Braunsteins and Hautphenne (2020) contains examples with four and five distinct extinction probability vectors.…”