Harnack inequality and interior regularity for Markov processes with degenerate jump kernels

TL;DR

This paper establishes a scale-invariant Harnack inequality and interior regularity results for purely discontinuous Markov processes with potentially degenerate jump kernels, advancing potential theory in stochastic processes.

Contribution

It introduces conditions under which the Harnack inequality holds for Markov processes with degenerate jump kernels and proves related interior regularity results.

Findings

Harnack inequality holds under certain conditions for degenerate jump kernels

Interior regularity of harmonic functions is established

A Dynkin-type formula for these processes is proved

Abstract

In this paper we study interior potential-theoretic properties of purely discontinuous Markov processes in proper open subsets $D\subset \mathbb{R}^d$. The jump kernels of the processes may be degenerate at the boundary in the sense that they may vanish or blow up at the boundary. Under certain natural conditions on the jump kernel, we show that the scale invariant Harnack inequality holds for any proper open subset $D\subset \mathbb{R}^d$ and prove some interior regularity of harmonic functions. We also prove a Dynkin-type formula and several other interior results.