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
Non-intersection exponents for Brownian paths
Download PDF
Download PDF
  • Published: September 1990

Non-intersection exponents for Brownian paths

Part I. Existence and an invariance principle

  • Krzysztof Burdzy1 &
  • Gregory F. Lawler2 

Probability Theory and Related Fields volume 84, pages 393–410 (1990)Cite this article

  • 141 Accesses

  • 25 Citations

  • Metrics details

Summary

LetX andY be independent 3-dimensional Brownian motions,X(0)=(0,0,0),Y(0)=(1,0,0) and letp r =P(X[0,r] ⋂Y[0,r]=∅). Then the “non-intersection exponent”\(\mathop {\lim }\limits_{r \to \infty } - {{\log p_{_r } } \mathord{\left/ {\vphantom {{\log p_{_r } } {\log r}}} \right. \kern-\nulldelimiterspace} {\log r}}\) exists and is equal to a similar “non-intersection exponent” for random walks. Analogous results hold inR 2 and for more than 2 paths.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Aizenman, M.: Geometric analysis ofφ 4 fields and Ising models. Parts I and II. Commun. Math. Phys.86, 1–48 (1982)

    Google Scholar 

  2. Burdzy, K., Lawler, G.: Non-intersection exponents for Brownian paths. Part II. Estimates and applications to a random fractal. Ann. Probab (to appear)

  3. Burdzy, K., Lawler, G., Polaski, T.: On the critical exponent for random walk intersections. J. Stat. Phys.56, 1–12 (1989)

    Google Scholar 

  4. Csörgö, M., Révész, P.: Strong approximations in probability and statistics. New York: Academic Press 1981

    Google Scholar 

  5. Doob, J.L.: Classical potential theory and its probabilistic counterpart. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  6. Duplantier, B., Kwon, K.-H.: Conformal invariance and intersections of random walks. Phys. Rev. Lett.61, 2514–2517 (1988)

    Google Scholar 

  7. Dvoretzky, A., Erdös, P., Kakutani, S.: Double points of paths of Brownian motion inn-space. Acta Sci. Math.12, 75–81 (1950)

    Google Scholar 

  8. Erdös, P., Taylor, S.J.: Some intersection properties of random walk paths. Acta Math. Sci. Hung.11, 231–248 (1960)

    Google Scholar 

  9. Lawler, G.: Intersections of random walks with random sets. Israel J. Math.65, 113–132 (1989)

    Google Scholar 

  10. Lawler, G.: Estimates for differences and Harnack's inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous increments. Proc. Lond. Math. Soc. (to appear)

  11. Meyer, P.A., Smythe, R.T., Walsh, J.B.: Birth and death of Markov processes. Proc. Sixth Berkeley Symp. Math. Stat. Prob, 1970/71, vol. III, pp. 295–305. Berkeley 1972

    Google Scholar 

  12. Port, S.C., Stone, C.J.: Brownian motion and classical potential theory. New York: Academic Press 1978

    Google Scholar 

  13. Williams, D.: Diffusions, Markov processes and martingales, vol. I. New York: Wiley 1979

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, University of Washington, GN-50, Seattle, WA, 98195, USA

    Krzysztof Burdzy

  2. Mathematics Department, Duke University, 27706, Durham, NC, USA

    Gregory F. Lawler

Authors
  1. Krzysztof Burdzy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gregory F. Lawler
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by NSF grant DMS 8702620

Supported by NSF grant DMS 8702879 and an Alfred P. Sloan Research Fellowship

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Burdzy, K., Lawler, G.F. Non-intersection exponents for Brownian paths. Probab. Th. Rel. Fields 84, 393–410 (1990). https://doi.org/10.1007/BF01197892

Download citation

  • Received: 17 December 1988

  • Revised: 17 July 1989

  • Issue Date: September 1990

  • DOI: https://doi.org/10.1007/BF01197892

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

  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Mathematical Biology
  • Analogous Result
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