Potential theory of Dirichlet forms degenerate at the boundary: the case of no killing potential

TL;DR

This paper studies the boundary behavior of jump processes with degenerate boundary conditions, establishing the boundary Harnack principle and Green function estimates for a class of Dirichlet forms without killing potential.

Contribution

It proves the boundary Harnack principle and sharp Green function estimates for jump processes with degenerate boundary behavior, extending previous results to cases with no killing potential.

Findings

Hunt process has finite lifetime and dies at the boundary for ta  (1,2)

Established boundary Harnack principle for degenerate jump processes

Derived sharp two-sided Green function estimates

Abstract

In this paper we consider the Dirichlet form on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-\alpha}\mathcal{B}(x,y)$, where $\mathcal{B}(x,y)$ can be degenerate at the boundary. Unlike our previous works [6,7] where we imposed critical killing, here we assume that the killing potential is identically zero. In case $\alpha\in (1,2)$ we first show that the corresponding Hunt process has finite lifetime and dies at the boundary. Then, as our main contribution, we prove the boundary Harnack principle and establish sharp two-sided Green function estimates. Our results cover the case of the censored $\alpha$-stable process, $\alpha\in (1,2)$, in the half-space studied in [2].