Skip to main content
SpringerLink
Log in

Harnack’s Inequality for Stable Lévy Processes

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We prove Harnack’s inequality for harmonic functions of a symmetric stable Lévy process on Rd without the assumption that the density function of its Lévy measure is locally bounded from below.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bass, R.F. and Kassmann, M.: ‘Harnack inequalities for non-local operators of variable order’, Trans. Amer. Math. Soc., to appear.

  2. Bass, R.F. and Levin, D.A.: ‘Harnack inequalities for jump processes’, Potential Anal. 17(4) (2002), 375–388.

    Google Scholar 

  3. Bertoin, J.: Lévy Processes, Cambridge University Press, Cambridge, 1996.

    Google Scholar 

  4. Blumenthal, R.M. and Getoor, R.K.: ‘Some theorems on stable processes’, Trans. Amer. Math. Soc. 95 (1960), 263–273.

    MATH  Google Scholar 

  5. Blumenthal, R.M. and Getoor, R.K.: Markov Processes and Potential Theory, Pure Appl. Math., Academic Press Inc., New York, 1968.

    Google Scholar 

  6. Blumenthal, R.M., Getoor, R.K. and Ray, D.B.: ‘On the distribution of first hits for the symmetric stable processes’, Trans. Amer. Math. Soc. 99 (1961), 540–554.

    Google Scholar 

  7. Bogdan, K.: ‘Sharp estimates for the Green function in Lipschitz domains’, J. Math. Anal. Appl. 243(2) (2000), 326–337.

    Google Scholar 

  8. Bogdan, K. and Byczkowski, T.: ‘Probabilistic proof of boundary Harnack principle for α-harmonic functions’, Potential Anal. 11 (1999), 135–156.

    Google Scholar 

  9. Bogdan, K. and Byczkowski, T.: ‘Potential theory of the Schrödinger operator based on the fractional Laplacian’, Probab. Math. Statist. 20 (2000), 293–335.

    Google Scholar 

  10. Bogdan, K., Stós, A. and Sztonyk, P.: ‘Potential theory for Lévy stable processes’, Bull. Polish Acad. Sci. Math. 50(3) (2002), 361–372.

    Google Scholar 

  11. Bogdan, K., Stós, A. and Sztonyk, P.: ‘Harnack inequality for symmetric stable processes on fractals’, C. R. Acad. Sci. Paris Math. 335(1) (2002), 59–63.

    Google Scholar 

  12. Bogdan, K., Stós, A. and Sztonyk, P.: ‘Harnack inequality for stable processes on d-sets’, Studia Math. 158(2) (2003), 168–198.

    Google Scholar 

  13. Chung, K.L. and Zhao, Z.: From Brownian Motion to Schrödinger’s Equation, Springer-Verlag, New York, 1995.

    Google Scholar 

  14. Dynkin, E.B.: Markov Processes, Vols. I, II, Springer, New York, 1965.

    Google Scholar 

  15. Dziubański, J.: ‘Asymptotic behaviour of densities of stable semigroups of measures’, Probab. Theory Related Fields 87 (1991), 459–467.

    Google Scholar 

  16. Getoor, R.K.: ‘First passage times for symmetric stable processes in space’, Trans. Amer. Math. Soc. 101 (1961), 75–90.

    Google Scholar 

  17. Głowacki, P. and Hebisch, W.: ‘Pointwise estimates for densities of stable semigroups of measures’, Studia Math. 104 (1993), 243–258.

    Google Scholar 

  18. Ikeda, N. and Watanabe, S.: ‘On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes’, J. Math. Kyoto Univ. 2(1) (1962), 79–95.

    Google Scholar 

  19. Jacob, N.: Pseudo-Differential Operators and Markov Processes, Vol. II: Generators and Their Potential Theory, Imperial College Press, London, 2002.

    Google Scholar 

  20. Jakubowski, T.: ‘The estimates for the Green function in Lipschitz domains for the symmetric stable processes’, Probab. Math. Statist. 22(2) (2002), 419–441.

    Google Scholar 

  21. Millar, P.W.: ‘First passage distributions of processes with independent increments’, Ann. Probab. 3(2) (1975), 215–233.

    Google Scholar 

  22. Moser, J.: ‘On Harnack’s theorem for elliptic differential equations’, Comm. Pure Appl. Math. 14 (1961), 577–591.

    Google Scholar 

  23. Polya, G.: ‘On the zeros of an integral function represented by Fourier’s integral’, Messenger of Math. 52 (1923), 185–188.

    Google Scholar 

  24. Safonov, M.: ‘Harnack’s inequality for elliptic equations and the Hölder property of their solutions’, J. Soviet Math. 21 (1983), 851–863.

    Google Scholar 

  25. Song, R. and Vondraček, Z.: ‘Harnack inequality for some classes of Markov processes’, Preprint.

  26. Sztonyk, P.: ‘On harmonic measures for Lévy processes’, Probab. Math. Statist. 20(2) (2000), 383–390.

    Google Scholar 

  27. Sztonyk, P.: ‘Boundary potential theory for stable processes’, Colloq. Math. 95(2) (2003), 191–206.

    Google Scholar 

  28. Taylor, S.J.: ‘Sample path properties of a transient stable process’, J. Math. Mech. 16 (1967), 1229–1246.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Krzysztof Bogdan & PaweŁ Sztonyk

  2. Institute of Mathematics, Polish Academy of Sciences, ul. Kopernika 18, 51-617, Wrocław, Poland

    Krzysztof Bogdan

Authors
  1. Krzysztof Bogdan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. PaweŁ Sztonyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Krzysztof Bogdan.

Additional information

Mathematics Subject Classifications (2000)

Primary 60J45, 31C05; Secondary 60G51.

Research partially supported by KBN (2P03A 041 22) and RTN (HPRN-CT-2001-00273-HARP).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bogdan, K., Sztonyk, P. Harnack’s Inequality for Stable Lévy Processes. Potential Anal 22, 133–150 (2005). https://doi.org/10.1007/s11118-004-0590-x

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-004-0590-x

Keywords

Search

Navigation