Invariance principle for random walks on dynamically averaging random conductances

TL;DR

This paper establishes an invariance principle for continuous-time random walks in environments where conductances dynamically average and stabilize over time, demonstrating convergence to standard behavior despite initial fluctuations.

Contribution

It introduces a novel invariance principle for random walks in environments with time-dependent conductance fluctuations that diminish under diffusive scaling.

Findings

Random walks converge to standard Brownian motion as conductances stabilize.

The environment's fluctuations diminish according to diffusive scaling.

Coupling with simple random walk proves the invariance principle.

Abstract

We prove an invariance principle for continuous-time random walks in a dynamically averaging environment on $\mathbb Z$. In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk.