A branching random walk among disasters

TL;DR

This paper studies a branching random walk in a random environment with disasters, providing criteria for survival, extinction, and growth rates of particles, extending previous models of random walks in hazardous environments.

Contribution

It introduces a new model of branching random walk among disasters and establishes criteria for survival and extinction, including growth rate analysis.

Findings

Positive survival probability criterion established

Almost sure extinction in the critical case proved

Exponential growth of particles shown when survival occurs

Abstract

We consider a branching random walk in a random space-time environment of disasters where each particle is killed when meeting a disaster. This extends the model of the "random walk in a disastrous random environment" introduced by [15]. We obtain a criterion for positive survival probability, see Theorem 1. The proofs for the subcritical and the supercritical cases follow standard arguments, which involve moment methods and a comparison with an embedded branching process with i.i.d. offspring distributions. The proof of almost sure extinction in the critical case is more difficult and uses the techniques from [8]. We also show that, in the case of survival, the number of particles grows exponentially fast.