Recurrence and Transience for Branching Random Walks in an I.I.D. Random Environment

#### Carl Mueller

2008, v.14, Issue 1, 115-130

ABSTRACT

We give three different criteria for transience of a branching Markov chain. These conditions enable us to give a classification of Branching Random Walks in Random Environment (BRWRE) on Cayley graphs in recurrence and transience. This classification is stated explicitly for BRWRE on $Z^d$. Furthermore, we emphasize the interplay between branching Markov chains and the spectral radius. We prove properties of the spectral radius of the Random Walk in Random Environment with the help of appropriate branching Markov chains.

Keywords: branching Markov chains,recurrence and transience,random walk in random environment,Cayley graph,spectral radius