Harnack Inequalities for some Lévy Processes

Abstract

In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝd. We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Levy process in ℝd with a Levy density given by \(c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}\), where 0 ≤ j(r) ≤ cr − d − α, ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion with subordinator with Laplace exponent ϕ(λ) = λα/2l(λ), λ > 0, where l is a slowly varying function at infinity and α ∈ (0,2).