Gradient estimates of the heat kernel for random walks among time-dependent random conductances

TL;DR

This paper establishes sharp heat kernel derivative estimates for random walks in time-dependent, ergodic random environments with unbounded conductances, leading to local limit theorems and Green function derivatives.

Contribution

It extends previous results to unbounded conductances with finite first moments using an adapted entropy method.

Findings

Sharp on-diagonal heat kernel derivative estimates

Local limit theorems for heat kernels

Extension to Green function derivatives under weak off-diagonal bounds

Abstract

In this paper we consider a time-continuous random walk in $\mathbb{Z}^d$ in a dynamical random environment with symmetric jump rates to nearest neighbours. We assume that these random conductances are stationary and ergodic and, moreover, that they are bounded from below but unbounded from above with finite first moment. We derive sharp on-diagonal estimates for the annealed first and second discrete space derivative of the heat kernel which then yield local limit theorems for the corresponding kernels. Assuming weak algebraic off-diagonal estimates, we then extend these results to the annealed Green function and its first and second derivative. Our proof which extends the result of Delmotte and Deuschel (2005) to unbounded conductances with first moment only, is an adaptation of the recent entropy method of Benjamini et. al. (2015).