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Dirichlet heat kernel estimates for parabolic nonlocal equations

Philipp Svinger, Marvin Weidner

TL;DR

This work develops a comprehensive boundary regularity theory for parabolic nonlocal equations in divergence form with Hölder continuous, time-dependent kernels on $C^{1,\alpha}$ domains. By combining kernel freezing, a parabolic Morrey–Campanato/Excess-decay framework, barrier methods, and tail analysis, it establishes the optimal $C^s_p$ boundary regularity and a quantitative boundary Harnack principle. It also proves a parabolic Hopf lemma and, crucially, the first two-sided Dirichlet heat kernel bounds for general time-dependent nonlocal operators in divergence form. The results extend known elliptic and time-homogeneous theories to fully parabolic, time-dependent kernels, with potential applications to nonlocal free boundary problems and stochastic processes with jumps.

Abstract

In this article we establish the optimal $C^s$ boundary regularity for solutions to nonlocal parabolic equations in divergence form in $C^{1,α}$ domains and prove a higher order boundary Harnack principle in this setting. Our approach applies to a broad class of nonlocal operators with merely Hölder 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.

Dirichlet heat kernel estimates for parabolic nonlocal equations

TL;DR

This work develops a comprehensive boundary regularity theory for parabolic nonlocal equations in divergence form with Hölder continuous, time-dependent kernels on domains. By combining kernel freezing, a parabolic Morrey–Campanato/Excess-decay framework, barrier methods, and tail analysis, it establishes the optimal boundary regularity and a quantitative boundary Harnack principle. It also proves a parabolic Hopf lemma and, crucially, the first two-sided Dirichlet heat kernel bounds for general time-dependent nonlocal operators in divergence form. The results extend known elliptic and time-homogeneous theories to fully parabolic, time-dependent kernels, with potential applications to nonlocal free boundary problems and stochastic processes with jumps.

Abstract

In this article we establish the optimal boundary regularity for solutions to nonlocal parabolic equations in divergence form in domains and prove a higher order boundary Harnack principle in this setting. Our approach applies to a broad class of nonlocal operators with merely Hölder 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.

Paper Structure

This paper contains 27 sections, 32 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Optimal boundary regularity
  3. Dirichlet heat kernel estimates
  4. Strategy of the proofs
  5. Freezing kernels and optimal regularity in the translation invariant case
  6. Optimal regularity for equations in divergence form
  7. Outline
  8. Acknowledgments
  9. Preliminaries
  10. Notation and weak solution concept
  11. Auxiliary results for weak solutions
  12. An iteration lemma
  13. Translation invariant case
  14. Regularity estimates via barrier functions
  15. Liouville theorem in half-space
  16. ...and 12 more sections

Key Result

Theorem 1

Let $s\in (0,1)$, $p >2$, $\alpha,\sigma \in (0,s)$, and $\Omega \subset \mathds{R}^d$ be a $C^{1,\alpha}$ domain with $0\in \partial \Omega$. Furthermore, let $\mathcal{L}_t$ be an operator of the form eq:OperatorDivergenceForm-eq:KernelDivergenceForm, satisfying eq:KernelHoelderCont with $\mathcal for some $f\in L^q_tL^r_x (Q_1^\Omega)$ where $q,r\geq 1$ satisfy $\frac{1}{q}+\frac{d}{2sr} <\frac

Theorems & Definitions (74)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Definition 5
  • Remark 6
  • Remark 7
  • Proposition 8
  • proof
  • Proposition 9
  • ...and 64 more