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 . We obtain a criterion for positive survival probability, see Theorem 1.1.
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. However, for this comparison we need to show that the survival rate of a single particle equals the survival rate of a single particle returning to the origin (Proposition 3.1). We prove this statement by making use of stochastic domination.
The proof of almost sure extinction in the critical case is more difficult and uses the techniques from , going back to . We also show that, in the case of survival, the number of particles grows exponentially fast.
"A branching random walk among disasters." Electron. J. Probab. 22 1 - 34, 2017. https://doi.org/10.1214/17-EJP75