Ergodic Theorems for Random Walks in Random Environments

TL;DR

This paper establishes ergodic theorems for random walks in stationary ergodic environments, proving a law of large numbers and invariance principles without requiring uniform ellipticity, thus broadening understanding of their long-term behavior.

Contribution

It introduces a general framework using Environment Functions and extends ergodic results to non-uniformly elliptic environments, including a uniqueness principle and transfer techniques.

Findings

Proves a law of large numbers for the random walk.

Establishes an invariance principle under minimal assumptions.

Shows transience behavior matches simple symmetric random walk.

Abstract

We study the Ergodic Properties of Random Walks in stationary ergodic environments without uniform ellipticity under a minimal assumption. There are two main components in our work. The first step is to adopt the arguments of Lawler to first prove a uniqueness principle. We use a more general definition of environments using~\textit{Environment Functions}. As a corollary, we can deduce an invariance principle under these assumptions for balanced environments under some assumptions. We also use the uniqueness principle to show that any balanced, elliptic random walk must have the same transience behaviour as the simple symmetric random walk. The second is to transfer the results we deduce in balanced environments to general ergodic environments(under some assumptions) using a control technique to derive a measure under which the \textit{local process} is stationary and ergodic. As a consequence of our results, we deduce the Law of Large Numbers for the Random Walk and an Invariance Principle under our assumptions.