Survival, extinction and approximation of discrete-time branching random walks

TL;DR

This paper studies the survival and extinction behavior of discrete-time branching random walks on countable sets, relating local and global survival to the first-moment matrix, and shows how these processes can be approximated by confined or multitype contact processes.

Contribution

It establishes a connection between survival properties and the first-moment matrix, and introduces approximation methods for locally surviving branching random walks.

Findings

Local survival is characterized by the first-moment matrix M.

Global survival cannot be fully described by M alone.

Locally surviving walks can be approximated by confined or multitype contact processes.

Abstract

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.