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Arkiv för Matematik

, Volume 48, Issue 2, pp 231–242 | Cite as

On hypoellipticity of generators of Lévy processes

  • Helmut AbelsEmail author
  • Ryad Husseini
Article
  • 59 Downloads

Abstract

We give a sufficient condition on a Lévy measure μ which ensures that the generator L of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., \(\mathop {\mathrm {sing\,supp}}u\subseteq \mathop {\mathrm {sing\,supp}}Lu\) for all admissible u. In particular, we assume that \(\mu|_{\mathbb {R}^{d}\setminus \{0\}}\in C^{\infty}(\mathbb {R}^{d}\setminus \{0\})\) . We also show that this condition is necessary provided that \(\mathop {\mathrm {supp}}\mu\) is compact.

Keywords

Harmonic Function Supporting Function Dirichlet Form Harnack Inequality Jump Process 
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Copyright information

© Institut Mittag-Leffler 2009

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Institute for Applied MathematicsUniversity of BonnBonnGermany

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