Invariance principles for rough walks in random conductances

TL;DR

This paper proves invariance principles for random walks in random conductances within the rough path topology, accommodating degenerate environments and long-range jumps, using a unified structural approach.

Contribution

It introduces a modular framework for pathwise convergence of random walks in complex environments, extending existing results to more general settings.

Findings

Established annealed and quenched invariance principles in rough path topology.

Provided conditions on the corrector's potential for convergence.

Developed a transfer lemma to construct the potential from spatial bounds.

Abstract

We establish annealed and quenched invariance principles for random walks in random conductances lifted to the p-variation rough path topology, allowing for degenerate environments and long-range jumps. Our proof is based on a unified structural strategy where pathwise convergence is viewed as a natural upgrade of the classical theory. This approach decouples the martingale lift from terms involving the integrals with respect to the corrector and the quadratic covariations. In the quenched regime, we show that the existence of a stationary potential for the corrector with $2+\epsilon$ moments is sufficient to ensure the vanishing of the corrector in $p$-variation for any $p>2$. This input, combined with our structural framework, provides a direct and modular pathway to rough path convergence. We further provide a transfer lemma to construct this potential from spatial moment bounds. While presently verified in the literature primarily for nearest-neighbor settings, our formulation isolates the exact analytic input required for pathwise convergence in more general environments.