Heat kernel estimates for Dirichlet forms degenerate at the boundary

TL;DR

This paper derives sharp two-sided heat kernel estimates for symmetric Markov processes with boundary-degenerate jump kernels in the upper half-space, revealing new behaviors and factorization properties influenced by boundary conditions.

Contribution

It provides the first sharp two-sided heat kernel estimates for non-local operators with boundary-degenerate jump kernels, including processes killed by potentials or boundary hitting.

Findings

Heat kernel estimates vary with parameter regions, showing three different forms.

Establishes approximate factorization of heat kernels with survival probabilities depending on boundary distance.

Reveals new qualitative behaviors not previously known for such non-local operators.

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 $J(x,y)=\mathcal B(x,y)|x-y|^{-\alpha-d}$, $\alpha\in (0,2)$, 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.