Potential theory of Dirichlet forms with jump kernels blowing up at the boundary

TL;DR

This paper investigates the potential theory of jump processes with boundary-blowing kernels and killing potentials, establishing boundary Harnack principles and sharp Green function estimates for various parameters.

Contribution

It extends potential theory to jump processes with singular boundary behaviors, proving boundary Harnack principles and Green function estimates in this setting.

Findings

Boundary Harnack principle holds for all admissible parameters.

Sharp two-sided Green function estimates are established.

Results apply to jump kernels with boundary blow-up and various dimensions.

Abstract

In this paper we study the potential theory of Dirichlet forms on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-\alpha}\mathcal{B}(x,y)$ and the killing potential $\kappa x_d^{-\alpha}$, where $\alpha\in (0, 2)$ and $\mathcal{B}(x,y)$ can blow up to infinity at the boundary. The jump kernel and the killing potential depend on several parameters. For all admissible values of the parameters involved and all $d \ge 1$, we prove that the boundary Harnack principle holds, and establish sharp two-sided estimates on the Green functions of these processes.