Potential Analysis

, Volume 39, Issue 1, pp 1–11

Continuity of Harmonic Functions for Non-local Markov Generators

Authors

    • Mathematical InstituteTohoku University
Article

DOI: 10.1007/s11118-012-9319-4

Cite this article as:
Wada, M. Potential Anal (2013) 39: 1. doi:10.1007/s11118-012-9319-4

Abstract

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).

Keywords

A priori estimatesContinuity of harmonic functionsMarkov jump processesMartingale problem

Mathematics Subject Classifications (2010)

45K0531B0535B6560J75

Copyright information

© Springer Science+Business Media Dordrecht 2012