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
Boundary trace of reflecting Brownian motions
Download PDF
Download PDF
  • Published: 02 February 2004

Boundary trace of reflecting Brownian motions

  • Itai Benjamini1,
  • Zhen-Qing Chen2,3 &
  • Steffen Rohde2,4 

Probability Theory and Related Fields volume 129, pages 1–17 (2004)Cite this article

  • 181 Accesses

  • 9 Citations

  • Metrics details

Abstract.

We establish a uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of non-smooth domains. Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are determined. Extensions to stable-like jump processes and to symmetric reflecting diffusions are also given.

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. Bass, R.F, Burdzy, K., Chen, Z.-Q.: Uniqueness for reflecting Brownian motion in lip domains. Preprint, 2002

  3. Bass, R.F, Hsu, P.: Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 19, 486–508 (1991)

    MathSciNet  MATH  Google Scholar 

  4. Bogdan, K., Burdzy, K., Chen, Z.-Q.: Censored stable processes. Probab. Theory Relat. Fields 127, 89–152 (2003)

    MATH  Google Scholar 

  5. Burdzy, K., Chen, Z.-Q.: Coalesence of synchronous couplings. Probab. Theory Relat. Fields 123, 553–578 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Z.-Q.: On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Relat. Fields. 94, 281–351 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Stochastic Process Appl. 108, 27–62 (2003)

    Article  Google Scholar 

  8. Davies, E.B: Heat Kernels and Spectral Theory. Cambridge Univ. Press, 1989

  9. Fabes, E.B, Stroock, D.W: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Ration. Mech. Anal. 96, 327–338 (1986)

    MathSciNet  MATH  Google Scholar 

  10. Fukushima, M.: A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math. 4, 183-215 (1967)

    MATH  Google Scholar 

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

  12. Fukushima, M., Uemura, T.: On Sobolev and capacitary inequalities for contractive Besov spaces over d-sets. Potential Anal. 18, 59–77 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hawkes, J.: Some dimension theorems for the sample function of stable processes. Indiana Univ. Math. J. 20, 733-738 (1971)

    MATH  Google Scholar 

  14. Hawkes, J.: On the Hausdorff dimension of the intersection of the range of a stable process with a Borel set. Z. Wahrsch. 19, 90–102 (1971)

    MATH  Google Scholar 

  15. Herron, D.A, Koskela, P.: Uniform, Sobolev extension and quasiconformal circle domains. J. D’Anal. Math. 57, 172–202 (1991)

    MathSciNet  MATH  Google Scholar 

  16. Jacob, N., Schilling, R.: Some Dirichlet spaces obtained by subordinate reflected diffusions. Rev. Mat. Iberoamericana 15, 59–91 (1999)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. Jerison, D., Kenig, C.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46, 80–147 (1982)

    MathSciNet  MATH  Google Scholar 

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

  20. Kaufman, R.: Une propriété métrique du mouvement brownien. R. Cad. Sc. Paris 268, 727–728 (1969)

    MATH  Google Scholar 

  21. Kumagai, T.: Some remarks for jump processes on fractals. In: Grabner, P. and Woess,~W. (eds.), Trends in Math.: Fractals in Graz 2001, Birkhäuser, 2002, pp. 185–196

  22. Liu, L., Xiao, Y.: Hausdorff dimension theorem for self-similar Markov processes. Probab. Math. Statist. 18, 369–383 (1998)

    MathSciNet  MATH  Google Scholar 

  23. Makarov, N.: Fine structure of Harmonic measure. Algebra i Analiz 10 (1998), 1–62; translation in St. Petersburg Math. J. 10, 217–268 (1999)

    MathSciNet  Google Scholar 

  24. Maz’ja, V.G: Sobolev Spaces. Springer, Berlin-Heidelberg, 1985

  25. Peres, Y.: Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177, 417–434 (1996)

    MathSciNet  MATH  Google Scholar 

  26. Pruitt, W.E: Some dimension results for processes with independent increments. In: Stochastic Processes and Related Topics, Vol. 1, M.L. Puri, (ed.), Academic Press, 1975

  27. Pruitt, W.E, Taylor, S.J: Sample path properties of processes with stable components. Z. Wahrsch. verw. Gebiete 12, 267–289 (1969)

    MATH  Google Scholar 

  28. Stroock, D.W: Diffusion semigroups corresponding to uniformly elliptic divergence form operators. Séminaire de Probabilités, XXII, 316-347, Lecture Notes in Math. 1321, Springer, Berlin, 1988

  29. Väisälä, J.: Uniform domains. Tohoku Math. J. 40, 101–118 (1988)

    MathSciNet  Google Scholar 

  30. Xiao, Y. Random fractals and Markov processes. Preprint, 2002

Download references

Author information

Authors and Affiliations

  1. Weizmann Institute, Rehovot, 76100, Israel

    Itai Benjamini

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

    Zhen-Qing Chen & Steffen Rohde

  3. This research is supported in part by NSF Grant, DMS-0303310, and a RRF grant from University of Washington

    Zhen-Qing Chen

  4. This research is supported in part by NSF Grant, DMS-0201435

    Steffen Rohde

Authors
  1. Itai Benjamini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen-Qing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Steffen Rohde
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Itai Benjamini.

Additional information

Mathematics Subject Classification (2000):Primary 60G17, 60J60, Secondary 28A80, 30C35, 60G52, 60J50

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Benjamini, I., Chen, ZQ. & Rohde, S. Boundary trace of reflecting Brownian motions. Probab. Theory Relat. Fields 129, 1–17 (2004). https://doi.org/10.1007/s00440-003-0318-7

Download citation

  • Received: 21 July 2003

  • Revised: 29 October 2003

  • Published: 02 February 2004

  • Issue Date: May 2004

  • DOI: https://doi.org/10.1007/s00440-003-0318-7

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

Key words or phrases

  • Reflecting Brownian motion
  • Hausdorff dimension
  • Uniform dimensional result
  • Boundary occupation time
  • Boundary trace
  • Conformal mapping
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