Dirichlet heat kernel estimates for parabolic nonlocal equations

TL;DR

This paper establishes optimal boundary regularity and heat kernel estimates for nonlocal parabolic equations with H"older continuous coefficients, including time-dependent cases, advancing understanding of their boundary behavior.

Contribution

It provides the first optimal boundary regularity results and sharp heat kernel estimates for nonlocal parabolic equations with time-dependent coefficients.

Findings

Optimal $C^s$ boundary regularity proved

Higher order boundary Harnack principle established

Sharp two-sided heat kernel estimates obtained

Abstract

In this article we establish the optimal $C^s$ boundary regularity for solutions to nonlocal parabolic equations in divergence form in $C^{1,\alpha}$ domains and prove a higher order boundary Harnack principle in this setting. Our approach applies to a broad class of nonlocal operators with merely H\"older continuous coefficients, but our results are new even in the translation invariant case. As an application, we obtain sharp two-sided estimates for the associated Dirichlet heat kernel. Notably, our estimates cover nonlocal operators with time-dependent coefficients, which had remained open in the literature.