Harnack inequalities for Hunt processes with Green function

Let (X,W) be a balayage space, 1 ∈ W, or – equivalently – let W be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that W separates points, every function in W is the supremum of its continuous minorants and there exist strictly positive continuous u, v ∈ W such that u/v → 0 at infinity. We suppose that there is a Green function G > 0 for X, a metric ρ on X and a decreasing function g : [0,∞) → (0,∞] having the doubling property and a mild upper decay near 0 such that G ≈ g ◦ ρ (which is equivalent to a 3G-inequality). Then the corresponding capacity for balls of radius r is bounded by a constant multiple of 1/g(r). Assuming that reverse inequalities hold as well and that jumps of the process, when starting at neighboring points, are related in a suitable way, it is proven that positive harmonic functions satisfy scaling invariant Harnack inequalities. Provided that the Ikeda-Watanabe formula holds, sufficient conditions for this relation are given. This shows that rather general Lévy processes are covered by this approach.