A generating function approach to branching random walks

Abstract

It is well known that the behaviour of a branching process is completely described by\ud the generating function of the offspring law and its fixed points. Branching random walks are a natural\ud generalization of branching processes: a branching process can be seen as a one-dimensional\ud branching random walk. We define a multidimensional generating function associated to a given\ud branching random walk. The present paper investigates the similarities and the differences of the\ud generating functions, their fixed points and the implications on the underlying stochastic process,\ud between the one-dimensional (branching process) and the multidimensional case (branching random\ud walk). In particular, we show that the generating function of a branching random walk can\ud have uncountably many fixed points and a fixed point may not be an extinction probability, even\ud in the irreducible case (extinction probabilities are always fixed points). Moreover, the generating\ud function might not be a convex function. We also study how the behaviour of a branching random\ud walk is affected by local modifications of the process. As a corollary, we describe a general\ud procedure by which we can modify a continuous-time branching random walk which has a weak\ud phase and turn it into a continuous-time branching random walk which has strong local survival\ud for large or small values of the parameter and non-strong local survival for intermediate values of\ud the parameter

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.…”

Global survival of branching random walks and tree-like branching random walks

“…where Ψ is the space of finitely supported functions in N X . This generating function has been introduced in [4, Section 3] (see also [6,9,26] for additional properties). In the case of the discretetime counterpart of a continuous-time BRW, the x-coordinate of G(z) can be written as…”

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

“…For A ⊆ X , we let q i (A) = P lim n→∞ ∈A Z n, = 0 ϕ 0 = i be the probability that there exists a finite generation after which we never see an individual with a type in the set A, given that the population begins with a single individual of type i, and we let q(A) := (q i (A)) i∈X be the corresponding extinction probability vector. The vectors q(A) are also elements of S (see (2)). Such a general definition of extinction leads to redundancies.…”

“…To generalise (i)-(iii) to the infinite type setting it is generally accepted that we should give the corresponding results for both q andq. That is, we aim to (i) derive a partial and a global extinction criterion, (ii) develop iterative methods to compute q andq when an algebraic expression cannot be found, and (iii) locate q andq in S. While open questions remain, a number of authors have made progress on (i) [7,9,15,18,20], (ii) [6,11,16], and (iii) [2,7,15] (to name a few).…”

“…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?…”