Survival of branching random walks in random environment

TL;DR

This paper investigates the survival behavior of branching random walks in random environments on the integer lattice, providing a spectral radius criterion for local survival and a Lyapunov exponent criterion for global survival.

Contribution

It generalizes the classification of BRWRE into recurrent and transient regimes and introduces a new characterization of global survival using Lyapunov exponents.

Findings

Local and strong local survival regimes coincide.

Spectral radius characterizes local survival.

Lyapunov exponents determine global survival.

Abstract

We study survival of nearest-neighbour branching random walks in random environment (BRWRE) on ${\mathbb Z}$. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. $2\times 2$ random matrices.