On the Satic and Dynamic Points of View for Certain Random Walks in Random Environment
Erwin Bolthausen, Alain-Sol Sznitman
2002
Methods and Applications of Analysis
In this work we prove the equivalence between static and dynamic points of views for certain ballistic random walks in random environment on Z Z d , when d ≥ 4 and the disorder is low. Our techniques also enable us to derive in the same setting a functional central limit theorem for almost every realization of the environment. We also provide an example where the equivalence between static and dynamic points of views breaks down. 0. Introduction. In many models of random motions in random
more »
... the "environment viewed from the particle" naturally defines a Markov chain. The existence of an invariant measure of this chain, which is absolutely continuous with respect to the "static" distribution of the environment is an important property. It is the starting point in the analysis of the "environment viewed from the particle", a technique which has been one of the key tools on the investigation of random motions in random media, c.f. Kipnis-Varadhan [10], S.M. Kozlov [11], Molchanov [13], Olla [14], Papanicolaou-Varadhan [15]. However this technique has had relatively little impact for one of the basic examples of random motions in random media, namely random walks in random environment. In particular, the question of the equivalence between the "static" and "dynamic" distributions of the environment is poorly understood in this situation, with the few exceptions of dimension one, cf. Kesten [9], Molchanov [13], p. 273-274, and of walks with null local drift, cf. Lawler [12], and Papanicolaou-Varadhan [15] in the continuous setting. The present work proves the equivalence between static and dynamic distributions of the environment, for certain ballistic random walks in random environment, in dimension d ≥ 4, when the disorder is low. The techniques we develop enable us to derive a "quenched" central limit theorem, which complements the results of Sznitman [18] . Let us now recall the model. The environment in which the walk evolves is described by a collection of i.i.d. (2d)-dimensional vectors, which specify the transition probability of the walk at each site of Z Z d . We assume that for some κ ∈ [0, 1 2d ], (0.1) the common law µ of the vectors is supported by P κ , the set of (2d)-vectors (p(e)) |e|=1,e∈Z Z d , with p(e) ∈ [κ, 1], for |e| = 1, and |e|=1 p(e) = 1 . Our principal interest lies in the elliptic situation, when (0.2) κ > 0 . However, the discussion of what we nickname "directed walks", will also be useful. It *
doi:10.4310/maa.2002.v9.n3.a4
fatcat:gk3wodkeercyfgrg274k7pg3wi