A generating function approach to branching random walks

TL;DR

This paper explores the properties of generating functions in multidimensional branching random walks, revealing complex behaviors such as multiple fixed points and non-convexity, and examines how local modifications influence the process.

Contribution

It introduces a multidimensional generating function framework for branching random walks and analyzes their fixed points, highlighting differences from classical branching processes.

Findings

Generating functions can have uncountably many fixed points.

Fixed points may not correspond to extinction probabilities.

Local modifications can alter the survival behavior of the process.

Abstract

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