Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation

TL;DR

This paper demonstrates that Dirichlet heat kernel estimates for a broad class of non-local Markov processes remain stable under non-local Feynman-Kac perturbations, extending their robustness in various geometric settings.

Contribution

It establishes the stability of heat kernel estimates for non-symmetric Markov processes under Feynman-Kac perturbations, including stable-like and censored processes in complex domains.

Findings

Heat kernel estimates are stable under perturbations.

Includes processes on closed $d$-sets and open $C^{1,1}$ sets.

Applicable to processes with drifts and censored processes.

Abstract

In this paper we show that Dirichlet heat kernel estimates for a class of (not necessarily symmetric) Markov processes are stable under non-local Feynman-Kac perturbations. This class of processes includes, among others, (reflected) symmetric stable-like processes on closed $d$-sets in $\bR^d$, killed symmetric stable processes, censored stable processes in $C^{1, 1}$ open sets as well as stable processes with drifts in bounded $C^{1, 1}$ open sets.