Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary

TL;DR

This paper derives sharp two-sided Green function estimates for a class of jump processes with boundary-degenerate kernels, identifying parameter regions where the boundary Harnack principle holds or fails.

Contribution

It provides the first comprehensive Green function estimates for these boundary-degenerate jump processes, clarifying when the boundary Harnack principle applies.

Findings

Sharp two-sided Green function estimates established.

Boundary Harnack principle holds in certain parameter regions.

Boundary Harnack principle fails in some other parameter regions.

Abstract

In this paper we continue our investigation of the potential theory of Markov processes with jump kernels decaying at the boundary. To be more precise, we consider processes in ${\mathbb R}^d_+$ with jump kernels of the form ${\mathcal B}(x,y) |x-y|^{-d-\alpha}$ and killing potentials $\kappa(x)=cx_d^{-\alpha}$, $0<\alpha<2$. The boundary part ${\mathcal B}(x,y)$ is comparable to the product of three terms with parameters $\beta_1, \beta_2$, $\beta_3$ and $\beta_4$ appearing as exponents in these terms. The constant $c$ in the killing term can be written as a function of $\alpha$, ${\mathcal B}$ and a parameter $p\in ((\alpha-1)_+, \alpha+\beta_1)$, which is strictly increasing in $p,$ decreasing to $0$ as $p\downarrow (\alpha-1)_+$ and increasing to $\infty$ as $p\uparrow\alpha+\beta_1$. We establish sharp two-sided estimates on the Green functions of these processes for all $p\in ((\alpha-1)_+, \alpha+\beta_1)$ and all admissible values of $\beta_1, \beta_2$, $\beta_3$ and $\beta_4$. Depending on the regions where $\beta_1$, $\beta_2$ and $p$ belong, the estimates on the Green functions are different. In fact, the estimates have three different forms depending on the regions the parameters belong to. As applications, we prove that the boundary Harnack principle holds in certain region of the parameters and fails in some other region of the parameters. Combined with the main results of \cite{KSV},we completely determine the region of the parameters where the boundary Harnack principle holds.