Critical parameters of germ-monotone families of branching random walks

TL;DR

This paper introduces a broad class of branching random walks parameterized by a positive parameter, exploring how their critical parameters change with modifications in reproduction laws, extending classical stochastic domination.

Contribution

It defines a general notion of critical parameters for germ-monotone families of branching random walks and analyzes their behavior under law modifications.

Findings

Introduction of germ order for branching random walks

Definition of critical parameters associated with subsets

Analysis of how reproduction law modifications affect critical parameters

Abstract

We introduce a broad class of families of branching random walks on a set $X$. The processes in each family are parametrized by a positive parameter $\lambda$ and they are monotonically increasing in $\lambda$ with respect to the germ order, a notion that extends classical stochastic domination. We define a general notion of critical parameter $\lambda(A)$ associated with a subset $A \subseteq X$ and investigate how modifications of reproduction laws affect these critical parameters.