Ergodic Theorems for Random Walks in Random Environments

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.

💡 Research Summary

The paper investigates ergodic properties of random walks in random environments (RWRE) without the usual uniform ellipticity assumption. Its main goal is to establish a Law of Large Numbers (LLN) and an Invariance Principle (functional central limit theorem) for a broad class of stationary ergodic environments, including those that are merely elliptic in a weak sense. The authors introduce the notion of an “environment function” E, which maps a configuration ω to transition probabilities at each lattice site. A balanced environment is defined by the symmetry condition E_i(ω)(x)=E_{i+d}(ω)(x) for all directions i, and ellipticity is required only in the sense that each transition probability is positive almost surely.

The first major contribution adapts Lawler’s 1982 arguments to this more general setting. By employing a discrete Monge‑Ampère operator M and a uniqueness principle for concave functions, the authors prove that for any measurable balanced environment function satisfying a mild moment condition
\