Abstract
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 Comets and Popov [3] 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.
Highlights
A branching Markov chain (BMC) is a system of particles in discrete time
We say a BMC is irreducible if for every starting position x and every state y there is a positive probability that y will be visited by some particle
In the second part we follow the line of research of Comets, Menshikov and Popov [2], Comets and Popov [3], Machado and Popov [6], [7] and the author [9] and study transience and recurrence of branching random walk in random environment (BRWRE)
Summary
A branching Markov chain (BMC) is a system of particles in discrete time. The BMC starts with one particle in an arbitrary starting position x. In the second part we follow the line of research of Comets, Menshikov and Popov [2], Comets and Popov [3], Machado and Popov [6], [7] and the author [9] and study transience and recurrence of branching random walk in random environment (BRWRE). In this case we can use the criterion of the first part, Theorem 2.4, to obtain a classification of BRWRE in transient and strong recurrent regimes, Theorem 3.2. The obtained classification result for BRWRE, Theorem 3.2, is quite interesting facing the difficulty of the corresponding questions for random walks in random environment of a single particle, compare with Sznitman [15] and [16]
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