Heat kernel estimates for Markov processes in bounded sets with jump kernels decaying at the boundary

TL;DR

This paper derives sharp two-sided heat kernel estimates for symmetric Markov processes with jump kernels decaying at the boundary in bounded smooth domains, considering both conservative and killed processes.

Contribution

It provides the first sharp two-sided heat kernel estimates for jump processes with boundary-decaying kernels in bounded smooth domains.

Findings

Established sharp two-sided heat kernel estimates in Lipschitz and $C^{1,1}$ domains.

Extended previous results to processes with boundary-decaying jump kernels.

Analyzed both conservative and killed Markov processes with boundary decay.

Abstract

In this paper, we study two types of purely discontinuous symmetric Markov processes $X$ in bounded smooth subsets of $\mathbb R^d$: conservative processes and processes killed either upon approaching the boundary of the set or by a killing potential $\kappa$. The jump kernel of $X$ is of the form $J(x,y)={\cal B}(x,y)|x-y|^{-d-\alpha}$, $\alpha\in (0,2)$, where the function ${\cal B}(x,y)$ decays to 0 at the boundary and is described in terms of two $O$-regularly varying functions and one slowly varying function. Under the conditions, introduced in \cite{CKSV24}, on ${\cal B}(x,y)$ and on the killing potential $\kappa$, we establish sharp two-sided estimates on the heat kernel of $X$: in Lipschitz sets when $X$ is conservative, and in $C^{1,1}$ open sets for the killed process.