Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Heat kernel estimates for boundary traces of reflected diffusions on uniform domains

Naotaka Kajino, Mathav Murugan

TL;DR

The paper develops a comprehensive framework for boundary traces of reflected diffusions on uniform domains within metric measure Dirichlet spaces that satisfy MD and sub-Gaussian HKE. It establishes sharp two-sided bounds for harmonic and elliptic measures, identifies the boundary trace Dirichlet form via the Doob--Naïm formula as a pure-jump form with jump kernel equal to the continuous Naïm kernel extended to the boundary, and proves stable-like heat kernel estimates for the boundary trace process. The results extend classical boundary operators such as the Dirichlet-to-Neumann map to a wide class of diffusions and domains, including Lipschitz, NTA, and fractal boundaries, and address both bounded and unbounded uniform domains through the construction of boundary elliptic measures at infinity. This provides a probabilistic and analytic bridge between interior local diffusions and nonlocal boundary operators, with implications for potential theory, inverse problems, and the analysis of jump processes on rough boundaries.

Abstract

We study the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of independent interest such as sharp two-sided estimates and the volume doubling property of the harmonic measure, the existence of a continuous extension of the Naïm kernel to the topological boundary, and the Doob--Naïm formula identifying the Dirichlet form of the boundary trace process as the pure-jump Dirichlet form whose jump kernel with respect to the harmonic measure is exactly (the continuous extension of) the Naïm kernel.

Heat kernel estimates for boundary traces of reflected diffusions on uniform domains

TL;DR

The paper develops a comprehensive framework for boundary traces of reflected diffusions on uniform domains within metric measure Dirichlet spaces that satisfy MD and sub-Gaussian HKE. It establishes sharp two-sided bounds for harmonic and elliptic measures, identifies the boundary trace Dirichlet form via the Doob--Naïm formula as a pure-jump form with jump kernel equal to the continuous Naïm kernel extended to the boundary, and proves stable-like heat kernel estimates for the boundary trace process. The results extend classical boundary operators such as the Dirichlet-to-Neumann map to a wide class of diffusions and domains, including Lipschitz, NTA, and fractal boundaries, and address both bounded and unbounded uniform domains through the construction of boundary elliptic measures at infinity. This provides a probabilistic and analytic bridge between interior local diffusions and nonlocal boundary operators, with implications for potential theory, inverse problems, and the analysis of jump processes on rough boundaries.

Abstract

We study the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of independent interest such as sharp two-sided estimates and the volume doubling property of the harmonic measure, the existence of a continuous extension of the Naïm kernel to the topological boundary, and the Doob--Naïm formula identifying the Dirichlet form of the boundary trace process as the pure-jump Dirichlet form whose jump kernel with respect to the harmonic measure is exactly (the continuous extension of) the Naïm kernel.
Paper Structure (27 sections, 61 theorems, 348 equations)

This paper contains 27 sections, 61 theorems, 348 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Summary of the setting and statement of the main results
  4. Preliminaries
  5. Metric doubling and volume doubling properties
  6. Uniform domains
  7. Regular Dirichlet space and symmetric Hunt process
  8. Sub-Gaussian heat kernel estimates
  9. Harmonic functions and the elliptic Harnack inequality
  10. Trace Dirichlet form
  11. Stable-like heat kernel estimates
  12. Capacity good measures and their corresponding PCAFs
  13. Green function, Martin kernel, and Naïm kernel
  14. Properties of Green function
  15. Boundary Harnack principle
  16. ...and 12 more sections

Key Result

Proposition 1.2

Assume that the part Dirichlet form $(\mathcal{E}^{U},\mathcal{F}^{0}(U))$ on $U$ is transient. Then for each $x_{0} \in U$, there exists a unique continuous function $\Theta^{U}_{x_{0}}\colon(\overline{U}\setminus\{x_{0}\})^{2}_{\operatorname{od}}\to(0,\infty)$, called the Naïm kernel of $U$ with b Moreover, there exist $c_{0},C_{1}\in(0,\infty)$ such that for any $x_{0}\in U$ and any $(\xi,\eta)

Theorems & Definitions (165)

  • Remark 1.1
  • Proposition 1.2: Part of Proposition \ref{['p:naim']}
  • Theorem 1.3: Theorem \ref{['t:hmeas']} and Corollary \ref{['c:asdouble']}
  • Proposition 1.4: Part of Proposition \ref{['p:emeas']}
  • Theorem 1.5: Doob--Naïm formula; Proposition \ref{['prop:bdry-trace-pure-jump']} and Theorem \ref{['t:dnformula']}
  • Theorem 1.6: Non-probabilistic part of Theorem \ref{['thm:shk-trace']}
  • Definition 2.1: Metric doubling property (MD)
  • Definition 2.2: Volume doubling property (VD)
  • Definition 2.3
  • Lemma 2.4: Hei
  • ...and 155 more