Random walks in random environments without ellipticity

TL;DR

This paper proves ergodicity of random walks in certain non-elliptic environments on Z^d, leading to a quenched invariance principle for doubly stochastic martingale walks.

Contribution

It establishes ergodicity under weaker conditions than ellipticity and derives a general quenched invariance principle for specific environments.

Findings

Ergodicity of the environment viewed from the particle

Validity of the quenched invariance principle in new settings

Extension of invariance principles to non-elliptic environments

Abstract

We consider random walks in random environments on Z^d. Under a transitivity hypothesis that is much weaker than the customary ellipticity condition, and assuming an absolutely continuous invariant measure on the space of the environments, we prove the ergodicity of the annealed process w.r.t. the dynamics "from the point of view of the particle". This implies in particular that the environment viewed from the particle is ergodic. As an example of application of this result, we give a general form of the quenched Invariance Principle for walks in doubly stochastic environments with zero local drift (martingale condition).