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Heat kernel for reflected diffusion and extension property on uniform domains

Mathav Murugan

Abstract

We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of Jones (Acta Math. 1981) states that uniform domains in Euclidean space are extension domains for Sobolev spaces. In this work, we obtain a similar extension property for metric spaces equipped with a Dirichlet form whose heat kernel satisfies a sub-Gaussian estimate. We introduce a scale-invariant version of this extension property and apply it to show that the reflected diffusion process on such a uniform domain inherits various properties from the ambient space, such as Harnack inequalities, cutoff energy inequality, and sub-Gaussian heat kernel bounds. In particular, our work extends Neumann heat kernel estimates of Gyrya and Saloff-Coste (Astérisque 2011) beyond the Gaussian space-time scaling. Furthermore, our estimates on the extension operator imply that the energy measure of the boundary of a uniform domain is always zero. This property of the energy measure is a broad generalization of Hino's result (PTRF 2013) that proves the vanishing of the energy measure on the outer square boundary of the standard Sierpiński carpet equipped with the self-similar Dirichlet form.

Heat kernel for reflected diffusion and extension property on uniform domains

Abstract

We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of Jones (Acta Math. 1981) states that uniform domains in Euclidean space are extension domains for Sobolev spaces. In this work, we obtain a similar extension property for metric spaces equipped with a Dirichlet form whose heat kernel satisfies a sub-Gaussian estimate. We introduce a scale-invariant version of this extension property and apply it to show that the reflected diffusion process on such a uniform domain inherits various properties from the ambient space, such as Harnack inequalities, cutoff energy inequality, and sub-Gaussian heat kernel bounds. In particular, our work extends Neumann heat kernel estimates of Gyrya and Saloff-Coste (Astérisque 2011) beyond the Gaussian space-time scaling. Furthermore, our estimates on the extension operator imply that the energy measure of the boundary of a uniform domain is always zero. This property of the energy measure is a broad generalization of Hino's result (PTRF 2013) that proves the vanishing of the energy measure on the outer square boundary of the standard Sierpiński carpet equipped with the self-similar Dirichlet form.
Paper Structure (26 sections, 27 theorems, 94 equations, 1 figure)

This paper contains 26 sections, 27 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Framework and Main results
  3. Doubling metric space and doubling measures
  4. Uniform domains
  5. Metric measure Dirichlet space and energy measure
  6. Sub-Gaussian heat kernel estimates
  7. Main results
  8. Outline of the work
  9. Whitney covering and uniform domains
  10. Whitney covering
  11. Basic properties of uniform domains
  12. Whitney cover on a uniform domain
  13. Reflection of Whitney balls à la Jones
  14. Sub-Gaussian heat kernel estimates
  15. Extension domain
  16. ...and 11 more sections

Key Result

Theorem 2.7

Let $(X,d,m,{\mathcal{E}},{\mathcal{F}})$ be an MMD space that satisfies the heat kernel estimate hke for some scale function $\Psi$ and let $m$ be a doubling measure. Let $U$ be a uniform domain $U$ and let $({\mathcal{E}}_U,{\mathcal{F}}(U))$ denote the bi-linear form in Definition d:local. There Here $\Gamma, \Gamma_U$ denote the energy measures of $({\mathcal{E}},{\mathcal{F}})$ and $({\mathc

Figures (1)

  • Figure 1: Outline of the work

Theorems & Definitions (65)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Local Dirichlet space and its energy measure
  • Definition 2.6: $\operatorname{ HKE(\Psi)}$
  • Theorem 2.7: Extension property
  • Theorem 2.8: Heat kernel estimate for reflected diffusion
  • Theorem 2.9: Energy measure of the boundary
  • Definition 3.1
  • ...and 55 more