Open Access
2017 A branching random walk among disasters
Nina Gantert, Stefan Junk
Electron. J. Probab. 22: 1-34 (2017). DOI: 10.1214/17-EJP75


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 1.1.

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 [8], going back to [1]. We also show that, in the case of survival, the number of particles grows exponentially fast.


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Nina Gantert. Stefan Junk. "A branching random walk among disasters." Electron. J. Probab. 22 1 - 34, 2017.


Received: 8 August 2016; Accepted: 12 June 2017; Published: 2017
First available in Project Euclid: 9 September 2017

zbMATH: 1379.60100
MathSciNet: MR3698736
Digital Object Identifier: 10.1214/17-EJP75

Primary: 60J80 , 60K37 , 82D60

Keywords: Branching random walk , random environment , survival

Vol.22 • 2017
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