Stability and instability of Gaussian heat kernel estimates for random walks among time-dependent conductances

TL;DR

This paper investigates the stability of Gaussian heat kernel estimates for time-dependent random walks with conductances, revealing instability in discrete and continuous cases under certain conditions, and stability under specific site-dependent holding times.

Contribution

It demonstrates that Gaussian heat kernel estimates are generally unstable for time-dependent conductances, but can be stable when holding times are site-dependent and uniform.

Findings

Two-sided Gaussian heat kernel estimates are not stable under perturbations in discrete time.

Instability occurs for continuous-time random walks with i.i.d. exponential holding times.

Stability is achieved when holding times vary by sites with a uniform base measure.

Abstract

We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is proved by giving an example of a ballistic and transient time-dependent random walk on Z among uniformly elliptic time-dependent conductances. For continuous time random walks, we show the instability when the holding times are i.i.d. exp(1), and in contrast, we prove the stability when the holding times change by sites in such a way that the base measure is a uniform measure.