Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extension method in Dirichlet spaces with sub-Gaussian estimates and applications to regularity of jump processes on fractals

Fabrice Baudoin, Quanjun Lang, Yannick Sire

TL;DR

The paper addresses regularity for non-local equations $(-L)^s f=0$ on regular, strongly local Dirichlet spaces that satisfy sub-Gaussian heat kernel bounds. It develops a general extension method on the augmented space $X_a=X\times\mathbb{R}$ with the operator $L_a=L+\mathcal{B}_a$ (where $a=1-2s$) to relate $s$-harmonic extensions to the original fractional operator via a Poisson representation and a Dirichlet-to-Neumann formula. By proving Hölder continuity and a Harnack inequality for weak solutions through an anisotropic parabolic theory on the extended space and a trace argument, the work provides a cohesive extension framework in Dirichlet spaces, including fractal settings. It also discusses Sobolev–Besov function spaces for existence results and outlines boundary Harnack principles, highlighting open problems and connections to jump processes on fractals.

Abstract

We investigate regularity properties of some non-local equations defined on Dirichlet spaces equipped with sub-gaussian estimates for the heat kernel associated to the generator. We prove that weak solutions for homogeneous equations involving pure powers of the generator are actually Hölder continuous and satisfy an Harnack inequality. Our methods are based on a version of the Caffarelli-Silvestre extension method which is valid in any Dirichlet space and our results complement the existing literature on solutions of PDEs on classes of Dirichlet spaces such as fractals.

Extension method in Dirichlet spaces with sub-Gaussian estimates and applications to regularity of jump processes on fractals

TL;DR

The paper addresses regularity for non-local equations on regular, strongly local Dirichlet spaces that satisfy sub-Gaussian heat kernel bounds. It develops a general extension method on the augmented space with the operator (where ) to relate -harmonic extensions to the original fractional operator via a Poisson representation and a Dirichlet-to-Neumann formula. By proving Hölder continuity and a Harnack inequality for weak solutions through an anisotropic parabolic theory on the extended space and a trace argument, the work provides a cohesive extension framework in Dirichlet spaces, including fractal settings. It also discusses Sobolev–Besov function spaces for existence results and outlines boundary Harnack principles, highlighting open problems and connections to jump processes on fractals.

Abstract

We investigate regularity properties of some non-local equations defined on Dirichlet spaces equipped with sub-gaussian estimates for the heat kernel associated to the generator. We prove that weak solutions for homogeneous equations involving pure powers of the generator are actually Hölder continuous and satisfy an Harnack inequality. Our methods are based on a version of the Caffarelli-Silvestre extension method which is valid in any Dirichlet space and our results complement the existing literature on solutions of PDEs on classes of Dirichlet spaces such as fractals.
Paper Structure (20 sections, 13 theorems, 160 equations, 3 figures)

This paper contains 20 sections, 13 theorems, 160 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Dirichlet Spaces
  3. Basic definitions on Dirichlet spaces
  4. Tensorization of Dirichlet spaces
  5. Weak solutions associated to generators
  6. Sub-Gaussian heat kernel estimate
  7. Caloric functions
  8. Extension Theorem on Dirichlet Spaces
  9. Pure Fractional powers
  10. Sketch of the argument for Theorem \ref{['thm:extension_theorem']}
  11. Extended Dirichlet space and tensorization
  12. Sub-Gaussian estimates and the existence of weak solutions
  13. Regularity of weak solutions and Harnack inequalities
  14. Weak local lower estimate
  15. Oscillation inequality and Hölder continuity
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose the sub-Gaussian heat kernel estimates above eq:subGaussian-intro hold on the Dirichlet space $(X,d,\mu,\mathcal{E}, \mathcal{F})$. Consider a non-negative weak solution $f$ of $(-L)^sf = 0$ where $-L$ is the generator of $\mathcal{E}$ and the parameter range is $s \in (0,1)$. Then the follo for any $x_0\in X$ and $R > 0$. Moreover, the following Hölder estimate holds, for $\mu$-almost al

Figures (3)

  • Figure 1: A part of an infinite, or unbounded, Vicsek set.
  • Figure 2: A part of an infinite, or unbounded, Sierpinski gasket.
  • Figure 3: A part of an infinite, or unbounded, Sierpinski carpet.

Theorems & Definitions (36)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 26 more