## Electronic Journal of Probability

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

### A branching random walk among disasters

#### 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 1.1.

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

**Source**

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

**Dates**

Received: 8 August 2016

Accepted: 12 June 2017

First available in Project Euclid: 9 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1504922530

**Digital Object Identifier**

doi:10.1214/17-EJP75

**Mathematical Reviews number (MathSciNet)**

MR3698736

**Zentralblatt MATH identifier**

1379.60100

**Subjects**

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

**Keywords**

branching random walk random environment survival

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

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