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A criterion for transience of multidimensional branching random walk in random environment. (English) Zbl 1191.60124

Summary: We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that Condition L of F. Comets and S. Popov [Ann. Probab. 35, No. 1, 68–114 (2007; Zbl 1114.60084)] is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branching random walks and implies that the critical branching random walk is transient resp. dies out locally.

MSC:

60K37 Processes in random environments
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Citations:

Zbl 1114.60084
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