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

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