Survival of branching random walks in random environment

Details Survival of branching random walks in random environment Survival of branching random walks in random environment

2008-11-11

arxiv.org/abs/0811.1748v3

Journal of Theoretical Probability, v. 23, p. 1002-1014, 2010

Mathematics

Probability

17 pages; to appear in Journal of Theoretical Probability

Scientific paper

10.1007/s10959-009-0227-5

We study survival of nearest-neighbour branching random walks in random environment (BRWRE) on ${\mathbb Z}$. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. $2\times 2$ random matrices.

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