Random Walk in Dynamic Markovian Random Environment

Abstract

We consider a model, introduced by Boldrighini, Minlos and Pellegrinotti, of random walks in dynamical random environments on the integer lattice Z^d with d>=1. In this model, the environment changes over time in a Markovian manner, independently across sites, while the walker uses the environment at its current location in order to make the next transition. In contrast with the cluster expansions approach of Boldrighini, Minlos and Pellegrinotti, we follow a probabilistic argument based on regeneration times. We prove an annealed SLLN and invariance principle for any dimension, and provide a quenched invariance principle for dimension d > 7, providing for d>7 an alternative to the analytical approach of Boldrighini, Minlos and Pellegrinotti, with the added benefit that it is valid under weaker assumptions. The quenched results use, in addition to the regeneration times already mentioned, a technique introduced by Bolthausen and Sznitman.