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Heat kernel estimates for Dirichlet forms degenerate at the boundary

Soobin Cho, Panki Kim, Renming Song, Zoran Vondraček

TL;DR

This paper develops sharp two-sided heat kernel estimates for purely discontinuous symmetric Markov processes in the upper half-space with jump kernels that degenerate at the boundary. By combining Nash-type inequalities, parabolic Hölder regularity, parabolic Harnack inequalities, and careful space-time analysis, the authors capture both one-jump and two-jump contributions, yielding upper and lower bounds that reflect boundary degeneracy. For killed processes, they establish approximate factorization of the heat kernel with survival probabilities tied to the distance to the boundary, and they obtain precise Green function estimates revealing a phase transition at a critical parameter value. The results extend nonlocal Dirichlet form theory to boundary-degenerate kernels and provide a detailed, technically rich picture of boundary effects on heat propagation and potential theory in half-spaces.

Abstract

The goal of this paper is to establish sharp two-sided estimates on the heat kernels of two types of purely discontinuous symmetric Markov processes in the upper half-space of $\mathbb R^d$ with jump kernels degenerate at the boundary. The jump kernels are of the form $J(x,y)=\mathcal B(x,y)|x-y|^{-α-d}$, $α\in (0,2)$, where the function $\mathcal B$ depends on four parameters and may vanish at the boundary. Our results are the first sharp two-sided estimates for the heat kernels of non-local operators with jump kernels degenerate at the boundary. The first type of processes are conservative Markov processes on $\overline{\mathbb R}^d_+$ with jump kernel $J(x,y)$. Depending on the regions where the parameters belong, the heat kernels estimates have three different forms, two of them are qualitatively different from all previously known heat kernel estimates. The second type of processes are the processes above killed either by a critical potential or upon hitting the boundary of the half-space. We establish that their heat kernel estimates have the approximate factorization property with survival probabilities decaying as a power of the distance to the boundary, where the power depends on the constant in the critical potential.

Heat kernel estimates for Dirichlet forms degenerate at the boundary

TL;DR

This paper develops sharp two-sided heat kernel estimates for purely discontinuous symmetric Markov processes in the upper half-space with jump kernels that degenerate at the boundary. By combining Nash-type inequalities, parabolic Hölder regularity, parabolic Harnack inequalities, and careful space-time analysis, the authors capture both one-jump and two-jump contributions, yielding upper and lower bounds that reflect boundary degeneracy. For killed processes, they establish approximate factorization of the heat kernel with survival probabilities tied to the distance to the boundary, and they obtain precise Green function estimates revealing a phase transition at a critical parameter value. The results extend nonlocal Dirichlet form theory to boundary-degenerate kernels and provide a detailed, technically rich picture of boundary effects on heat propagation and potential theory in half-spaces.

Abstract

The goal of this paper is to establish sharp two-sided estimates on the heat kernels of two types of purely discontinuous symmetric Markov processes in the upper half-space of with jump kernels degenerate at the boundary. The jump kernels are of the form , , where the function depends on four parameters and may vanish at the boundary. Our results are the first sharp two-sided estimates for the heat kernels of non-local operators with jump kernels degenerate at the boundary. The first type of processes are conservative Markov processes on with jump kernel . Depending on the regions where the parameters belong, the heat kernels estimates have three different forms, two of them are qualitatively different from all previously known heat kernel estimates. The second type of processes are the processes above killed either by a critical potential or upon hitting the boundary of the half-space. We establish that their heat kernel estimates have the approximate factorization property with survival probabilities decaying as a power of the distance to the boundary, where the power depends on the constant in the critical potential.
Paper Structure (10 sections, 72 theorems, 352 equations, 1 figure, 1 table)

This paper contains 10 sections, 72 theorems, 352 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and main results
  2. Nash inequality and the existence of the heat kernel
  3. Parabolic Hölder regularity and consequences
  4. Parabolic Harnack inequality and preliminary lower bound of heat kernels
  5. Preliminary heat kernel upper bounds
  6. Sharp heat kernel lower bounds
  7. Sharp heat kernel upper bounds
  8. Proofs of Theorems \ref{['t:HKE']} and \ref{['t:HKE-kappa']}
  9. Green function estimates
  10. Appendix: Estimates of some integrals

Key Result

Theorem 1.2

Suppose that (A1), (A3) and (A4) hold. Then the process $\overline Y$ can be refined to be a conservative Feller process with strong Feller property starting from every point in ${\overline{ \mathbb R}}^d_+$ and has a jointly continuous heat kernel $\overline p:(0,\infty) \times {\overline{ \math (b) When $d \ge 2$, we have that for all $(t,x,y) \in (0,\infty) \times {\overline{ \mathbb R}}^d_+

Figures (1)

  • Figure 1: Dominant path from $x$ to $y$ (with $x_d\le y_d$) at time $t<(|x-y|/6)^\alpha$ when $\beta_2>\alpha+\beta_1$, $\beta_3=\beta_4=0$.

Theorems & Definitions (135)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 125 more