Law of large numbers for non-elliptic random walks in dynamic random environments

TL;DR

This paper establishes a law of large numbers for certain non-elliptic random walks in dynamic environments, using a generalized regeneration scheme under a mixing condition, with applications to specific one-dimensional cases.

Contribution

It extends the law of large numbers to non-elliptic dynamic random environments using a novel regeneration approach and mixing assumptions.

Findings

Law of large numbers proven for non-elliptic random walks

Sign of the speed determined in some one-dimensional cases

Applicable to a broad class of dynamic environments

Abstract

We prove a law of large numbers for a class of $\Z^d$-valued random walks in dynamic random environments, including non-elliptic examples. We assume for the random environment a mixing property called \emph{conditional cone-mixing} and that the random walk tends to stay inside wide enough space-time cones. The proof is based on a generalization of a regeneration scheme developed by Comets and Zeitouni for static random environments and adapted by Avena, den Hollander and Redig to dynamic random environments. A number of one-dimensional examples are given. In some cases, the sign of the speed can be determined.