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

Abstract

We study variable-speed random walks on $\mathbb{Z} $ driven by a family of nearest-neighbor time-dependent random conductances $\{a_{t}(x,x+1)\colon x\in \mathbb{Z} ,\,t\ge 0\}$ 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.