Potential Analysis

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

Continuity of Harmonic Functions for Non-local Markov Generators

Article

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 estimates Continuity of harmonic functions Markov jump processes Martingale problem 

Mathematics Subject Classifications (2010)

45K05 31B05 35B65 60J75 

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References

  1. 1.
    Barlow, M.T., Bass, R.F., Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361(4), 1963–1999 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bass, R.F., Levin, D.A.: Harnack inequalities for jump processes. Potential Anal. 17, 375–388 (2002)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bass, R.F., Kassmann, M.: Harnack inequalities for non-local operators of variable order. Trans. Am. Math. Soc. 357(2), 837–850 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bass, R.F., Kassmann, M.: Hölder continuity of harmonic functions with respect to operators of variable order. Commun. Part. Diff. Eq. 30, 1249–1259 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Stoch. Proc. Appl. 108, 27–62 (2003)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Husseini, R., Kassmann, M.: Jump processes, \(\mathcal{L}\)-harmonic functions, continuity estimates and the Feller property. Ann. Inst. Henri Poincaré, B Calc. Probab. Stat. 45(4), 1099–1115 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Kassmann, M., Mimica, A.: Analysis of jump processes with nondegenerate jumping kernels. arXiv:1109.3678v2 [math. PR] 21 (2011)
  8. 8.
    Song, R., Vondraček, Z.: Harnack inequality for some classes of Markov processes. Math. Z. 246, 177–202 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversityAobaJapan

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