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

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.

A few typos corrected, some references updated; Accepted in the Journal of Functional Analysis