Potential Analysis

, Volume 39, Issue 1, pp 1–11 | Cite as

Continuity of Harmonic Functions for Non-local Markov Generators



In this paper, we treat a priori estimates of harmonic functions for jump processes associated with non-local operators. Let \(\mathcal{L}\) be a non-local operator given by
$$\mathcal{L}u(x) = \int_{\mathbb{R}^{d} \backslash \{0\}}(u(x+h)-u(x)-h \cdot \nabla u(x) 1_{\{|h| \leq 1\}}) n(x,h)dh.$$
Under some conditions on n(x,h), we prove the Hölder continuity and the uniform continuity of \(\mathcal{L}\)-harmonic functions. Our results are extensions of those obtained by Bass and Kassmann (Commun Part Diff Eq 30:1249–1259, 2005).


A priori estimates Continuity of harmonic functions Markov jump processes Martingale problem 

Mathematics Subject Classifications (2010)

45K05 31B05 35B65 60J75 


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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversityAobaJapan

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