Random walks among time increasing conductances: heat kernel estimates

TL;DR

This paper establishes Gaussian heat kernel estimates and Harnack inequalities for continuous-time random walks on graphs with time-increasing conductances, under certain regularity conditions.

Contribution

It provides the first two-sided heat kernel bounds and Harnack inequalities for random walks with time-varying conductances that increase over time.

Findings

Gaussian transition density estimates are proven.

Parabolic Harnack inequality is established.

Results apply to discrete-time lazy walks with matching bounds.

Abstract

For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks evolving via time varying, uniformly elliptic conductances, provided the vertex conductances (i.e. reversing measures), increase in time. Such transition density upper bounds apply for discrete time uniformly lazy walks, with the matching lower bounds holding once the parabolic Harnack inequality is proved.