Stochastic integrability of heat-kernel bounds for random walks in a balanced random environment

TL;DR

This paper establishes exponential integrability of heat kernel bounds and proves optimal diffusive decay for random walks in balanced i.i.d. environments, leading to a functional CLT for the environment viewed from the particle.

Contribution

It introduces stochastic integrability results for heat kernels and demonstrates diffusive decay, advancing understanding of random walks in balanced environments.

Findings

Exponential integrability of heat kernel bounds.

Optimal diffusive decay in dimensions d≥3.

Functional central limit theorem for the environment viewed from the particle.

Abstract

We consider random walks in a balanced i.i.d. random environment in $Z^d$ for $d\ge2$ and the corresponding discrete non-divergence form difference operators. We first obtain an exponential integrability of the heat kernel bounds. We then prove the optimal diffusive decay of the semigroup generated by the heat kernel for $d\ge3$. As a consequence, we deduce a functional central limit theorem for the environment viewed from the particle.