Strong Local Survival of Branching Random Walks is Not Monotone

Abstract

Abstract. The aim of this paper is the study of the strong local survival property for discretetime 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.

Extinction in lower Hessenberg branching processes with countably many types

“…At least partly due to these challenges, the set S is yet to be fully characterised in the infinite-type setting. There is, however, a number of papers that make progress toward this goal: Moyal [30] gives general conditions for S to contain at most a single solution s such that sup i∈X s i < 1; Spataru [36] gives a stronger results by stating that S contains at most two elements, q and 1; however, Bertacchi and Zucca [8,9] prove the inaccuracy of the latter by providing an irreducible example where S contains uncountably many elements such that sup i∈X s i = 1. Both q andq are elements of the set S. It is well known that q is the minimal element, but as yet, there has been no attempt to identify the precise location ofq.…”

“…To resolve the problem in the infinite-type setting we should give both a partial and a global extinction criterion. A number of authors have progressed in this direction [8,12,21,22,30,36,37]. In the infinite-type case, the analogue of the Perron-Frobenius eigenvalue is the convergence norm ν(M ) of M defined in (2.4), which gives a partial extinction criterion:q = 1 if and only if ν(M ) ≤ 1, see [37,Theorem 4.1].…”

“…It turns out that there can be no global extinction criterion based solely upon M , as highlighted through [37,Example 4.4], but as pointed out by the author, other moment conditions have not been clearly identified. In addition, wheñ q < 1, following the terminology in [8], the process can exhibit strong local survival q =q < 1, or non-strong local survival q <q < 1. It is again challenging to derive a general criterion separating the two cases.…”

On laws of large numbers in $L^{2}$ for supercritical branching Markov processes beyond $\lambda $-positivity

“…for every x ∈ J and all B ∈ B with ν(B) > 0, which is the analogue in our context of the notion of strong local survival studied in [7] and other references therein. However, in general it will not be equivalent to the concept of (plain) local survival introduced in [21], which is said to take place whenever there exists a compact set K ⊆ J such that…”

“…We show that, whenever (1) holds in L 2 , the equality P (D ∞ > 0|survival) = 1 is equivalent to the process ξ being strongly supercritical: this means that, in the event of survival, particles of the process can never accumulate all together on the boundary of the state space. This notion of strong supercriticality is related to the concept of strong local survival studied in [7] and references mentioned therein, although we are not aware of any previous connections made between this and the strict positivity of D ∞ on survival.…”