Potential Analysis

, Volume 25, Issue 1, pp 1–27

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

Article

DOI: 10.1007/s11118-005-9003-z

Cite this article as:
Rao, M., Song, R. & Vondraček, Z. Potential Anal (2006) 25: 1. doi:10.1007/s11118-005-9003-z

Abstract

Let X be a Lévy process in\(\mathbb{R}^{d} \), \(d \geqslant 3\), obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.

Key words

Subordinate Brownian motion Green function Harnack inequality subordinator complete Bernstein function capacity 

Mathematics Subject Classifications (2000)

Primary 60J45, 60J75 Secondary 60J25 

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleU.S.A.
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaU.S.A.
  3. 3.Department of MathematicsUniversity of ZagrebZagrebCroatia

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