Scaling invariant Harnack inequalities in a general setting

In a setting, where only “exit measures” are given, as they are associated with an arbitrary right continuous strong Markov process on a separable metric space, we provide simple criteria for the validity of Harnack inequalities for positive harmonic functions. These inequalities are scaling invariant with respect to a metric on the state space which, having an associated Green function, may be adapted to the special situation. In many cases, this also implies continuity of harmonic functions and Hölder continuity of bounded harmonic functions. The results apply to large classes of Lévy (and similar) processes.