Heat kernel estimates on local and non-local Dirichlet spaces satisfying a weak chain condition

TL;DR

This paper derives heat kernel estimates for diffusions and jump processes on metric measure spaces with a weak chain condition, improving existing results and illustrating practical validity through examples.

Contribution

It provides explicit heat kernel estimates for jump processes with scale differences and demonstrates the applicability of the weak chain condition in real scenarios.

Findings

Explicit heat kernel estimates for diffusions and jump processes.

Improved bounds for jump processes with different scales.

Validation of the weak chain condition through practical examples.

Abstract

In this paper, we focus on the heat kernel estimates for diffusions and jump processes on metric measure spaces satisfying a weak chain condition, where the length of a nearly shortest $\varepsilon$-chain between two points $x,y$ is comparable with a function of $d(x,y)$ and $\varepsilon$. For a diffusion, the best estimate is already given by Grigor'yan and Telcs, and we make it explicit in our particular case. For jump processes, especially those where the scale of the process is different with that of the jump kernel, we improve the results by Bae, Kang, Kim and Lee. Uniformity of the coefficients (or parameters) in the known estimates and metric transforms play the key role in our proof. We also show by examples how the weak chain condition is valid in practice.