Scaling invariant Harnack inequalities in a general setting

Abstract

A general setting is considered where only "exit measures" are given, as they are associated with arbitrary right continuous strong Markov process. Simple criteria for the validity of scaling invariant Harnack inequalities for positive harmonic functions are provided.

This implies continuity of harmonic functions and Hölder continuity of bounded harmonic functions. The results apply to large classes of Lévy processes.

Keywords

• Harnack inequality
• Hölder continuity
• Levy process
• Green function
• Krylov-Safonov estimate
• Ikeda-Watanabe formula