Branching random walks with uncountably many extinction probability vectors

TL;DR

This paper investigates the set of extinction probability vectors in branching random walks, revealing conditions for their diversity and constructing examples with uncountably many such vectors, advancing understanding of extinction behaviors.

Contribution

The paper introduces new criteria for distinguishing extinction probabilities across different sets and constructs examples with uncountably many extinction probability vectors.

Findings

Most cases have only two extinction probability vectors.

Explicit examples with finitely many vectors are known.

Constructed examples with uncountably many vectors.

Abstract

Given a branching random walk on a set $X$, we study its extinction probability vectors $\mathbf q(\cdot,A)$. Their components are the probability that the process goes extinct in a fixed $A\subseteq X$, when starting from a vertex $x\in X$. The set of extinction probability vectors (obtained letting $A$ vary among all subsets of $X$) is a subset of the set of the fixed points of the generating function of the branching random walk. In particular here we are interested in the cardinality of the set of extinction probability vectors. We prove results which allow to understand whether the probability of extinction in a set $A$ is different from the one of extinction in another set $B$. In many cases there are only two possible extinction probability vectors and so far, in more complicated examples, only a finite number of distinct extinction probability vectors had been explicitly found. Whether a branching random walk could have an infinite number of distinct extinction probability vectors was not known. We apply our results to construct examples of branching random walks with uncountably many distinct extinction probability vectors.