Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Censored stable processes
Download PDF
Download PDF
  • Published: 22 May 2003

Censored stable processes

  • Krzysztof Bogdan1,
  • Krzysztof Burdzy2 &
  • Zhen-Qing Chen2 

Probability Theory and Related Fields volume 127, pages 89–152 (2003)Cite this article

  • 1138 Accesses

  • 191 Citations

  • Metrics details

Abstract.

We present several constructions of a ``censored stable process'' in an open set D⊂R n, i.e., a symmetric stable process which is not allowed to jump outside D. We address the question of whether the process will approach the boundary of D in a finite time – we give sharp conditions for such approach in terms of the stability index α and the ``thickness'' of the boundary. As a corollary, new results are obtained concerning Besov spaces on non-smooth domains, including the critical exponent case. We also study the decay rate of the corresponding harmonic functions which vanish on a part of the boundary. We derive a boundary Harnack principle in C 1,1 open sets.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Adams, D.R., Hedberg, L.I.: Function spaces and potential theory. Springer, 1996

  2. Ancona, A.: Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann. Inst. Fourier (Grenoble) 28, 169–213 (1978)

    MathSciNet  MATH  Google Scholar 

  3. Bass, R., Burdzy, K.: A probabilistic proof of the boundary Harnack principle. In seminar on stochastic processes, 1989, Progr. Probab. 18, 1–16 (1990), Birkhäuser

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

  5. Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, 1968

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

    MATH  Google Scholar 

  7. Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. Studia Math. 123(1), 43–80 (1997)

    MATH  Google Scholar 

  8. Bogdan, K., Byczkowski, T.: Probabilistic proof of boundary Harnack principle for α-harmonic functions. Potential Anal. 11, 135–156 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bogdan, K., Byczkowski, T.: Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Statist. 20(2), 293–335 (2000)

    MATH  Google Scholar 

  10. Bogdan, K., Byczkowski, T.: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133(1), 53–92 (1999)

    MATH  Google Scholar 

  11. Burdzy, K., Chen, Z.-Q., Sylvester, J.: Heat equation and reflected Brownian motion in time-dependent domains. To appear in Ann. Probab. 2003

  12. Caetano, A.M.: Approximation by functions of compact support in Besov-Triebel- Lizorkin spaces on irregular domains. Studia Math. 142, 47–63 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Chen, Z.-Q.: Multidimensional symmetric stable processes. Korean J. Comput. Appl. Math. 6(2), 227–266 (1999)

    MATH  Google Scholar 

  14. Chen, Z.-Q.: On reflected Dirichlet spaces. Probab. Theory Relat. Fields 94, 135–162 (1992)

    MathSciNet  MATH  Google Scholar 

  15. Chen, Z.-Q., Fitzsimmons, P.J., Takeda, M., Ying, J., Zhang, T.-S.: Absolute continuity of symmetric Markov processes. To appear in Ann. Probab. 2003

  16. Chen, Z.-Q., Kim, P.: Green function estimate for censored stable processes. Probab. Theory Relat. Fields 124, 595–610 (2002)

    Article  Google Scholar 

  17. Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Preprints, 2002

  18. Chen, Z.-Q., Song, R.: Estimates on Green functions and Poisson kernels of symmetric stable processes. Math. Ann. 312, 465–601 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, Z.-Q., Song, R.: A note on Green function estimates for symmetric stable processes. Preprint, 2003

  20. Chen, Z.-Q., Song, R.: Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains. Illinois J. Math. 44(1), 138–160 (2000)

    MATH  Google Scholar 

  21. Chen, Z.-Q., Song, R.: Martin boundary and integral representation for harmonic functions of symmetric stable processes. J. Funct. Anal. 159, 267–294 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, Z.-Q., Song, R.: Intrinsic ultracontractivity and conditional gauge for symmetric stable processes. J. Funct. Anal. 150, 204–239 (1997)

    Article  MATH  Google Scholar 

  23. Chung, K.L.: Green's function for a ball. In: Seminar on Stochastic Processes, 1986, pages 1–13, Boston, 1987. Progr. Probab. Statist., 13, Birkhäuser.

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

  25. Dahlberg, B.: Estimates of harmonic measure. Arch. Rat. Mech. Anal. 65, 275–288 (1977)

    MATH  Google Scholar 

  26. Ethier, S.N., Kurtz, T.G.: Markov Processes. Characterization and Convergence. Wiley, New York, 1986

  27. Farkas, W., Jacob, N.: Sobolev spaces on non-smooth domains and Dirichlet forms related to subordinate reflecting diffusions. Math. Nachr. 224, 75–104 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Friedman, A.: Foundations of modern analysis. Dover Publ. Inc., New York, 1982

  29. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin, 1994

  30. Fukushima, M., Uemura, T.: On Sobolev and capacitary inequalities for contractive Besov spaces over d-sets. To appear in Potential Analysis, 2003

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

    Google Scholar 

  32. Gong, G., Qian, M.-P., Zhao, Z.: Killed diffusions and their conditioning. Probab. Theory Relat. Fields 80, 151–167 (1988)

    MathSciNet  MATH  Google Scholar 

  33. Hoh, W.: Pseudo differential operators with negative definite symbols of variable order. Rev. Mat. Iberoamericana 16(2), 219–241 (2000)

    MATH  Google Scholar 

  34. Ikeda, N., Nagasawa, M., Watanabe, S.: A construction of Markov process by piecing out. Proc. Japan Academy 42, 370–375 (1966)

    MATH  Google Scholar 

  35. Ikeda, N., 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), 79–95 (1962)

    MATH  Google Scholar 

  36. Jerison, D., Kenig, C.: Boundary value problems on Lipschitz domains. In: Studies in Partial Differential Equations, ed. W. Littman, Math. Assoc. Amer., 1982

  37. Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147, 71–88 (1981)

    MathSciNet  MATH  Google Scholar 

  38. Jonsson, A., Wallin, H.: Function Spaces on Subsets of R n. Math. Reports, Vol.2, Part 1, Harwood Acad. Publ., London, 1984

  39. Komatsu, T.: Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32, 833–860 (1995)

    Google Scholar 

  40. Kulczycki, T.: Properties of Green function of symmetric stable processes. Probab. Math. Statist. 17(2), 339–364 (1997)

    MATH  Google Scholar 

  41. Landkof, N.S.: Foundations of Modern Potential Theory. Springer-Verlag, New York, 1972

  42. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York-London, 1994

  43. Sharpe, M.: General Theory of Markov Processes. Academic Press, Inc. 1988

  44. Silverstein, M.L.: Symmetric Markov Processes. Lecture Notes in Math. vol. 426, Springer-Verlag, 1974

  45. Song, R., Wu, J.M.: Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168(2), 403–427 (1999)

    Article  MATH  Google Scholar 

  46. Triebel, H.: Interpolation theory, function spaces, differential operators, second edition. Johann Ambrosius Barth, 1995

  47. Wu, J.M.G.: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Ann. Inst. Fourier (Grenoble) 28(4), 147–167 (1978)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Krzysztof Bogdan

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195-4350, USA

    Krzysztof Burdzy & Zhen-Qing Chen

Authors
  1. Krzysztof Bogdan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Krzysztof Burdzy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhen-Qing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Krzysztof Bogdan.

Additional information

Research partially supported by NSF Grant DMS-0071486.

Mathematics Subject Classification (2000): Primary 60G52, Secondary 60G17, 60J45

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bogdan, K., Burdzy, K. & Chen, ZQ. Censored stable processes. Probab. Theory Relat. Fields 127, 89–152 (2003). https://doi.org/10.1007/s00440-003-0275-1

Download citation

  • Received: 20 November 2001

  • Revised: 18 March 2003

  • Published: 22 May 2003

  • Issue Date: September 2003

  • DOI: https://doi.org/10.1007/s00440-003-0275-1

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Decay Rate
  • Harmonic Function
  • Finite Time
  • Critical Exponent
  • Besov Space
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature