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 Lévy process in ℝd with a Lévy 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) = λ α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2).
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Research supported in part by the MZOS Grant 037-0372790-2801 of the Republic of Croatia.
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Mimica, A. Harnack Inequalities for some Lévy Processes. Potential Anal 32, 275–303 (2010). https://doi.org/10.1007/s11118-009-9153-5
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DOI: https://doi.org/10.1007/s11118-009-9153-5
Keywords
- Harnack inequality
- Random walk
- Green function
- Poisson kernel
- Stable process
- Harmonic function
- Subordinator
- Subordinate Brownian motion