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Sub-Gaussian heat kernel estimates for reflected diffusion on inner uniform domains

Riku Anttila

TL;DR

The paper addresses when sub-Gaussian heat kernel estimates, encoded by a scale function \\(\\Psi \\) and exponent \\(\\beta \\geq 2, \\) for a diffusion on a metric measure space, are inherited by the reflected diffusion on an inner uniform domain. It develops a Dirichlet-form approach, replacing the cutoff Sobolev inequality with a cutoff energy condition and employing Väisälä’s geometric result to transfer energy control to the domain; regularity and energy-measure alignment are established to connect the ambient and reflected forms. The main contribution proves that the reflected Dirichlet form on the path-metric completion \\(\\widetilde{\\Omega})\\ inherits the sub-Gaussian HK estimates from the ambient form, under doubling, geodesicity, and inner uniformity assumptions. This work broadens the applicability of sub-Gaussian diffusion theory to reflected processes on irregular domains within general metric measure spaces, providing a robust framework for extending HK theory beyond uniform domains.

Abstract

We prove that sub-Gaussian heat kernel estimates are inherited from a diffusion process on the ambient space to the reflected diffusion process on a subset which is an inner uniform domain.

Sub-Gaussian heat kernel estimates for reflected diffusion on inner uniform domains

TL;DR

The paper addresses when sub-Gaussian heat kernel estimates, encoded by a scale function \ and exponent \ for a diffusion on a metric measure space, are inherited by the reflected diffusion on an inner uniform domain. It develops a Dirichlet-form approach, replacing the cutoff Sobolev inequality with a cutoff energy condition and employing Väisälä’s geometric result to transfer energy control to the domain; regularity and energy-measure alignment are established to connect the ambient and reflected forms. The main contribution proves that the reflected Dirichlet form on the path-metric completion \\(\\widetilde{\\Omega})\\ inherits the sub-Gaussian HK estimates from the ambient form, under doubling, geodesicity, and inner uniformity assumptions. This work broadens the applicability of sub-Gaussian diffusion theory to reflected processes on irregular domains within general metric measure spaces, providing a robust framework for extending HK theory beyond uniform domains.

Abstract

We prove that sub-Gaussian heat kernel estimates are inherited from a diffusion process on the ambient space to the reflected diffusion process on a subset which is an inner uniform domain.

Paper Structure

This paper contains 10 sections, 15 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Metric spaces
  4. Inner uniform domains
  5. Dirichlet forms
  6. Heat kernel estimates
  7. Auxiliary lemmas
  8. Main theorem
  9. Regularity
  10. Proof of the main theorem

Key Result

Theorem 1.2

Let $(X,d,\mu)$ be a metric measure space where $(X,d)$ is complete and geodesic, and $\mu$ is doubling measure on $(X,d)$. Let $(\mathcal{E},\mathcal{F})$ be a strongly local regular Dirichlet form on $L^2(X,\mu)$ satisfying eq:HKMain for $\beta \geq 2$, $\Omega \subseteq X$ be an inner uniform dom

Theorems & Definitions (47)

  • Theorem 1.2: Theorem \ref{['thm:MainMain']}
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.12
  • Definition 2.13
  • ...and 37 more