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
Distoriton of area and conditioned Brownian motion
Download PDF
Download PDF
  • Published: September 1993

Distoriton of area and conditioned Brownian motion

  • P. S. Griffin1,
  • G. C. Verchota1 &
  • A. L. Vogel1 

Probability Theory and Related Fields volume 96, pages 385–413 (1993)Cite this article

  • 68 Accesses

  • 4 Citations

  • Metrics details

Summary

In a simply connected planar domainD the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic γ induces a decomposition ofD into disjoint subregions\(\Omega _j \mathop \cup \limits_j \Omega _j = D\) and that the subregions are obtained in a natural way using Euclidean geometric quantities relating γ toD. The lifetime associated with γ on each Ω j is then shown to be bounded by the product of the diameter of the smallest ball containing γ⋂Ω j and the diameter of the largest ball in Ω j . Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in Ω j it leads to finite lifetime estimates in a variety of domains of infinite area.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Ahlfors, L.V.: Conformal invariants. New York: McGraw-Hill 1973

    Google Scholar 

  2. Baernstein, A.: Integral means, univalent functions and circular symmetrization. Acta. Math.133, 139–169 (1974)

    Google Scholar 

  3. Bañuelos, R.: Intrinsic ultracontractivity and eigenfunction estimates for Schrodinger operators. J. Funct. Anal.99 (1991)

  4. Bañuelos, R.: Lifetime and heat kernel estimates in non-smooth domains. (to appear 1991)

  5. Bañuelos, R., Carroll, R.: Conditioned brownian motion and hyperbolic geodesics in simply connected domains. (Preprint 1991)

  6. Bañuelos, R., Davis, B.: A geometrical characterization of intrinsic ultracontractivity for planar domains with boundaries given by the graphs of functions. (Preprint 1992)

  7. Bass, R.F., Burdzy, K.: Lifetimes of conditoned diffusions. (Preprint 1991)

  8. Brelot, M., Choquet, G.: Espaces et lignes de green. Ann. Inst. Fourier3, 199–263 (1951)

    Google Scholar 

  9. Churchhill, R.V.: Fourier series and boundary value problems. New York: McGraw-Hill 1969

    Google Scholar 

  10. Cranston, M., McConnell, T.R.: The lifetime of conditioned brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 1–11 (1983)

    Google Scholar 

  11. Griffin, P.S., McConnell T.R., Verchota, G.C.: Conditioned brownian motion in simply connected planar domains. Ann. Inst. Henri Poincaré (to appear)

  12. Hayman, W.K.: Subharmonic functions, vol. 2. Lond. Math. Soc. Monogr.20 (1989)

  13. Helms, L.: Introduction to potential theory. New York: Interscience 1969

    Google Scholar 

  14. Jones, P.W.: Extension theorems for bmo. Indiana Math. J.29 41–66 (1980)

    Google Scholar 

  15. Pommerenke, Chr.: Univalent functions. Göttingen: Vandenhoeck and Ruprecht 1975

    Google Scholar 

  16. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press 1970

    Google Scholar 

  17. Xu, J.: The lifetime of conditioned brownian motion in planar domains of infinite area. Probab. Theory Relat. Fields87, 469–487 (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Syracuse University, 13244, Syracuse, NY, USA

    P. S. Griffin, G. C. Verchota & A. L. Vogel

Authors
  1. P. S. Griffin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. C. Verchota
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. L. Vogel
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research of the first author was supported in part by NSF Grant DMS-9100811

Research of the second author was supported in part by NSF Grant DMS-9105407

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Griffin, P.S., Verchota, G.C. & Vogel, A.L. Distoriton of area and conditioned Brownian motion. Probab. Th. Rel. Fields 96, 385–413 (1993). https://doi.org/10.1007/BF01292679

Download citation

  • Received: 04 August 1992

  • Revised: 26 February 1993

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01292679

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

Mathematics Subject Classification (1991)

  • 60J65
  • 31A15
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