Stationary Measures for Random Walks in a Random Environment with Random Scenery

. Let \Gamma act on a countable set V with only finitely many orbits. Given a \Gamma-invariant random environment for a Markov chain on V and a random scenery, we exhibit, under certain conditions, an equivalent stationary measure for the environment and scenery from the viewpoint of the random walker. Such theorems have been very useful in investigations of percolation on quasi-transitive graphs. Contents 1. Introduction 107 2. Definitions 110 3. Proofs 111 References 113 1. Introduction Given a state space for a Markov chain, one might assign transition probabilities randomly in order to finish specifying the Markov chain. In such a case, one speaks about random walk in a random environment , or RWRE for short. If we do not condition on the transition probabilities, such a stochastic process is usually no longer a Markov chain. The first investigation of RWREs is due to Solomon [9]. Their properties are often surprising. Alternatively, given a completely specified Markov chain, whi..