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A branching random walk among disasters (1608.02440v4)
Published 8 Aug 2016 in math.PR
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.