Asymptotic behaviour for random walks in random environments

Abstract

In this paper we consider limit theorems for a random walk in a random environment, (X n ). Known results (recurrence-transience criteria, law of large numbers) in the case of independent environments are naturally extended to the case where the environments are only supposed to be stationary and ergodic. Furthermore, if ‘the fluctuations of the random transition probabilities around are small’, we show that there exists an invariant probability measure for ‘the environments seen from the position of (X n )’. In the case of uniquely ergodic (therefore non-independent) environments, this measure exists as soon as (X n ) is transient so that the ‘slow diffusion phenomenon’ does not appear as it does in the independent case. Thus, under regularity conditions, we prove that, in this case, the random walk satisfies a central limit theorem for any fixed environment.