Gradient estimates of the heat kernel for random walks among time-dependent random conductances
Jean-Dominique Deuschel, Takashi Kumagai, Martin Slowik
TL;DR
The paper addresses gradient estimates for the heat kernel of a time-inhomogeneous random walk in a dynamical random conductance model with unbounded coefficients. It develops an entropy-based framework to bound spatial derivatives of the annealed heat kernel and derives sharp on-diagonal gradient estimates, leading to annealed local limit theorems and CLTs. Under weak off-diagonal control, these results extend to the annealed Green function and its first two derivatives, yielding robust gradient and potential estimates in time-dependent, degenerate environments. The findings advance understanding of diffusion in dynamic random media with minimal moment assumptions and demonstrate the efficacy of the entropy method in the unbounded-conductance setting, with implications for homogenization-type results and probabilistic potential theory in random media.
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).