An invariance principle for one-dimensional random walks in degenerate dynamical random environments

Abstract

We study random walks on the integers driven by a sample of time-dependent nearest-neighbor conductances that are bounded but are permitted to vanish over time intervals of positive Lebesgue-length. Assuming only ergodicity of the conductance law under space-time shifts and a moment assumption on the time to accumulate a unit conductance over a given edge, we prove that the walk scales, under a diffusive scaling of space and time, to a non-degenerate Brownian motion for a.e. realization of the environment. The conclusion particularly applies to random walks on one-dimensional dynamical percolation subject to fairly general stationary edge-flip dynamics.

Theorems & Definitions (16)

• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Theorem 2.2
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Proposition 3.4
• Lemma 3.5