Boundary Harnack inequality for Markov processes with jumps

TL;DR

This paper establishes a boundary Harnack inequality for jump-type Markov processes, providing a key tool for analyzing harmonic functions in complex stochastic systems with jumps across various spaces.

Contribution

It introduces a boundary Harnack inequality applicable to a wide class of jump processes on metric measure spaces, under specific kernel and domain conditions.

Findings

Valid for positive harmonic functions in arbitrary open sets

Applicable to subordinate Brownian motions and Lévy processes

Extends to jump processes on fractals and perturbed processes

Abstract

We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds for positive harmonic functions in arbitrary open sets. It applies, e.g., to many subordinate Brownian motions, L\'evy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schr\"odinger, drift and jump perturbations of such processes.