Davies method for heat kernel upper bounds of non-local Dirichlet forms on ultra-metric spaces

TL;DR

This paper uses the Davies method to efficiently derive upper bounds for heat kernels of non-local Dirichlet forms on ultra-metric spaces, leveraging the space's unique ultra-metric property.

Contribution

It introduces a novel application of the Davies method to ultra-metric spaces, revealing a new phenomenon where the heat kernel vanishes beyond a certain truncated range.

Findings

Heat kernel upper bounds are obtained using Davies method.

The heat kernel vanishes when points are separated by large enough balls.

The ultra-metric property is key to the new phenomenon.

Abstract

We apply the Davies method to give a quick proof for upper estimate of the heat kernel for the non-local Dirichlet form on the ultra-metric space. The key observation is that the heat kernel of the truncated Dirichlet form vanishes when two spatial points are separated by any ball of radius larger than the truncated range. This new phenomenon arises from the ultra-metric property of the space.