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Dirichlet heat kernel estimates for isotropic Levy processes with Gaussian components in Lipschitz open sets

Jie-Ming Wang

TL;DR

This work achieves the two-sided Dirichlet heat kernel estimates for isotropic Lévy processes with Gaussian components in Lipschitz open sets, revealing that Varopoulos-type estimates may fail off-diagonal in such domains. By developing a probabilistic framework based on the time-space process, parabolic Harnack inequalities, Carleson estimates, and careful analysis of exit times and survival probabilities, the authors obtain near-diagonal two-sided bounds and off-diagonal bounds that couple the global heat kernel with boundary survival. They establish necessary and sufficient conditions for Varopoulos-type Dirichlet heat kernel estimates, showing they are equivalent to a scale-invariant boundary Harnack principle for these processes. As an application, they derive explicit two-sided Dirichlet heat kernel estimates in $C^{1,1}$ domains, connecting Green functions, exit times, and heat kernels in a unified probabilistic framework. The results illuminate boundary behavior of nonlocal operators with Gaussian components and extend the scope of Varopoulos-type estimates beyond purely local or purely nonlocal settings.

Abstract

In this paper, the two-sided Dirichlet heat kernel estimates are obtained for a class of discontinuous isotropic Levy processes with Gaussian components in Lipschitz open sets. Furthermore, the necessary and sufficient conditions for the Varopoulos-type Dirichlet heat kernel estimates holding for such processes in Lipschitz open sets are derived.

Dirichlet heat kernel estimates for isotropic Levy processes with Gaussian components in Lipschitz open sets

TL;DR

This work achieves the two-sided Dirichlet heat kernel estimates for isotropic Lévy processes with Gaussian components in Lipschitz open sets, revealing that Varopoulos-type estimates may fail off-diagonal in such domains. By developing a probabilistic framework based on the time-space process, parabolic Harnack inequalities, Carleson estimates, and careful analysis of exit times and survival probabilities, the authors obtain near-diagonal two-sided bounds and off-diagonal bounds that couple the global heat kernel with boundary survival. They establish necessary and sufficient conditions for Varopoulos-type Dirichlet heat kernel estimates, showing they are equivalent to a scale-invariant boundary Harnack principle for these processes. As an application, they derive explicit two-sided Dirichlet heat kernel estimates in domains, connecting Green functions, exit times, and heat kernels in a unified probabilistic framework. The results illuminate boundary behavior of nonlocal operators with Gaussian components and extend the scope of Varopoulos-type estimates beyond purely local or purely nonlocal settings.

Abstract

In this paper, the two-sided Dirichlet heat kernel estimates are obtained for a class of discontinuous isotropic Levy processes with Gaussian components in Lipschitz open sets. Furthermore, the necessary and sufficient conditions for the Varopoulos-type Dirichlet heat kernel estimates holding for such processes in Lipschitz open sets are derived.

Paper Structure

This paper contains 8 sections, 28 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exit distribution and survival probability estimates
  4. Parabolic Carleson estimate and survival probability estimates
  5. Proof of Theorem \ref{['T2']}
  6. Upper bounds in Theorem \ref{['T2']}
  7. Lower bounds in Theorem \ref{['T2']}
  8. Proof of Theorem \ref{['T3']} and Corollary \ref{['C2']}

Key Result

Theorem 1.1

There are positive constants $C_k=C_k(d, \phi)$, $k=1, 2, 3$ such that for every $t>0$ and $x, y \in \mathbb R^d$, where with $\phi^{-1}$ being the inverse function of $\phi$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Proposition 2.2: Parabolic Harnack inequality
  • Lemma 2.3
  • Lemma 2.4
  • Definition 3.1
  • ...and 21 more