A branching random walk among disasters. (English) Zbl 1379.60100

Gantert, Nina; Junk, Stefan

Electron. J. Probab. 22, Paper No. 67, 34 p. (2017).

Summary: 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 T. Shiga [Probab. Theory Relat. Fields 108, No. 3, 417–439 (1997; Zbl 0884.60093)]. 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 of O. Garet and R. Marchand [Electron. Commun. Probab. 18, Paper No. 9, 15 p. (2013; Zbl 1306.60153)], going back to [C. Bezuidenhout and G. Grimmett, Ann. Probab. 18, No. 4, 1462–1482 (1990; Zbl 0718.60109)]. We also show that, in the case of survival, the number of particles grows exponentially fast.

MSC:

60K37	Processes in random environments
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
82D60	Statistical mechanics of polymers

Keywords:

branching random walk; random environment; survival

Citations:

Zbl 0884.60093; Zbl 1306.60153; Zbl 0718.60109