Heat kernel for reflected jump diffusion on Ahlfors regular domains

TL;DR

This paper develops heat kernel estimates for reflected jump diffusions on Ahlfors regular domains within metric measure spaces, providing a framework for understanding their behavior under certain capacity and Dirichlet form conditions.

Contribution

It introduces a method to extend Dirichlet forms from reflected domains to ambient spaces and establishes heat kernel estimates for reflected jump diffusions.

Findings

Construction of an extension operator with scale-invariant bounds

Derivation of mixed stable-like heat kernel estimates

Application to general metric measure spaces

Abstract

We study reflected jump diffusions on Ahlfors regular domains in general metric measure spaces. Under the condition that the Dirichlet form on the ambient space satisfies a capacity upper bound estimate, we construct an extension operator from the reflected Dirichlet space to the ambient Dirichlet space, with a scale-invariant local bound. Second, we establish the mixed stable-like heat kernel estimates for the reflected jump diffusion, assuming that the process on the ambient space satisfies the same type of heat kernel estimates.