General random walk in a random environment defined on Galton-Watson trees

Abstract

We consider a particle performing a random walk on a Galton-Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours are determined by a random process.We introduce a method for deriving conditions under which the walk is either transient or recurrent.We first suppose that the weights are i.i.d., and re-prove a result of Lyons and Pemantle (Ann. Probab. 20 (1992) 125-136). We then assume a Markovian environment along each line of descent, and finally consider a random walk in a Markovian environment that itself changes the environment. Our approach involves studying the typical behaviour of the walk on fixed lines of descent, which we then show determines the..