An invariance principle for one-dimensional random walks among dynamical random conductances

TL;DR

This paper establishes a quenched invariance principle for one-dimensional variable-speed random walks in time-dependent, ergodic random environments with minimal moment conditions, using a novel dual walk representation.

Contribution

It introduces a new approach to prove invariance principles under minimal assumptions by representing parabolic coordinates through a dual walk.

Findings

Proves a quenched invariance principle under minimal moment conditions.

Develops a novel dual walk representation for analysis.

Extends understanding of random walks in dynamic random environments.

Abstract

We study variable-speed random walks on $\mathbb Z$ driven by a family of nearest-neighbor time-dependent random conductances $\{a_t(x,x+1)\colon x\in\mathbb Z, t\ge0\}$ whose law is assumed invariant and ergodic under space-time shifts. We prove a quenched invariance principle for the random walk under the minimal moment conditions on the environment; namely, assuming only that the conductances possess the first positive and negative moments. A novel ingredient is the representation of the parabolic coordinates and the corrector via a dual random walk which is considerably easier to analyze.