Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances

TL;DR

This paper derives heat kernel upper bounds for continuous-time random walks with unbounded conductances, extending previous results to general speed measures and relating the intrinsic metric to the graph distance.

Contribution

It introduces a generalized framework for heat kernel estimates using intrinsic metrics induced by arbitrary speed measures under degenerate conductances.

Findings

Established heat kernel upper bounds governed by intrinsic metrics.

Extended previous results to a broader class of speed measures.

Provided a comparison between intrinsic metric and graph distance.

Abstract

We establish heat kernel upper bounds for a continuous-time random walk under unbounded conductances satisfying an integrability assumption, where we correct and extend recent results by the authors to a general class of speed measures. The resulting heat kernel estimates are governed by the intrinsic metric induced by the speed measure. We also provide a comparison result of this metric with the usual graph distance, which is optimal in the context of the random conductance model with ergodic conductances.