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

TL;DR

The paper addresses potential theory for purely discontinuous symmetric jump processes on the half-space with a boundary-blowing jump kernel $J(x,y)=|x-y|^{-d-\alpha}{\mathcal B}(x,y)$ and killing term $\kappa x_d^{-\alpha}$. It introduces a general resurrection (return) kernel to produce a symmetric, scale-invariant jump structure and proves a scale-invariant boundary Harnack principle with exact decay rate $p$, along with sharp two-sided Green function estimates across admissible parameters. The methodology combines Dynkin-type formulas, barrier constructions, Carleson estimates, Krylov-Safonov-type bounds, and interior Green function analysis, leveraging results from KSV22b and a detailed treatment of the resurrection kernel. The results provide a robust framework for boundary behavior of nonlocal Dirichlet forms with boundary blow-up, extending stability phenomena to boundary-enhanced jump kernels and enabling precise quantitative control of harmonic functions and Green potentials near 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 $J(x,y)=|x-y|^{-d-α}\mathcal{B}(x,y)$ and the killing potential $κx_d^{-α}$, where $α\in (0, 2)$ 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.

Theorems & Definitions (56)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Example 2.1
• Proposition 2.2
• Lemma 2.3
• Theorem 2.4
• Corollary 2.5
• Remark 2.6
• Lemma 3.1