Electronic Journal of Probability

A branching random walk among disasters

Nina Gantert and Stefan Junk

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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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 67, 34 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 82D60: Polymers

branching random walk random environment survival

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Gantert, Nina; Junk, Stefan. A branching random walk among disasters. Electron. J. Probab. 22 (2017), paper no. 67, 34 pp. doi:10.1214/17-EJP75. https://projecteuclid.org/euclid.ejp/1504922530

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