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
Asymptotics for the wiener sausage with drift
Download PDF
Download PDF
  • Published: March 1987

Asymptotics for the wiener sausage with drift

  • T. Eisele1 &
  • R. Lang1 

Probability Theory and Related Fields volume 74, pages 125–140 (1987)Cite this article

  • 139 Accesses

  • 28 Citations

  • Metrics details

Summary

A particle is considered which moves in ℝd according to a Brownian motion with drifth≠0. The space is assumed to contain random traps. The probability of survival of the particle up to timeT decays exponentially asT→∞ with a positive decay rate λ. λ is shown to be a non-analytic function of |h|. For small |h| the decay rate is given by λ(h)=1/2|h|2; but if |h| exceeds a certain critical value, λ(h) depends also on the parameters describing trapping. Upper and lower bounds for λ(h) are given, which imply the asymptotic linearity of λ(h) for large |h|. The critical point marks a transition from localized to delocalized behavior. A variational formula for the decay rate is given on the level of generalized processes, which elucidates the mathematical mechanism behind observations made earlier by Grassberger and Procaccia on the basis of computer simulations.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Comm. Pure Appl. Math.28, 525–565 (1975)

    Google Scholar 

  2. Grassberger, P., Procaccia, I.: Diffusion and drift in a medium with randomly distributed traps. Phys. Rev. A26, 3686–3688 (1982)

    Google Scholar 

  3. Hida, T.: Brownian motion. Berlin Heidelberg New York: Springer 1980

    Google Scholar 

  4. Kac, M.: Probability and related topics in physical sciences. London: Interscience-Wiley 1959

    Google Scholar 

  5. Kang, K., Redner, S.: Novel behavior of biased correlated walks in one dimension. J. Chem. Phys.80, 2752–2755 (1984)

    Google Scholar 

  6. Kusuoka, S.: The variational principle for stationary Gaussian Markov fields. In: Kallianpur, G. (ed.) Lect. Notes Control Inform.49, 179–187 (1983)

  7. Kusuoka, S.: Asymptotics of polymer measures in one dimension. In: Proc. Bielefeld meeting “Stochastic Processes and Infinite Dimensional Analysis” 1984

  8. Movaghar, B., Pohlmann, B., Wurtz, D.: Electric field dependence of tropping in one dimension. Phys. Rev. A29, 1568–1570 (1984)

    Google Scholar 

  9. Spitzer, F.: Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 110–121 (1964)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Im Neuenheimer Feld 294, D-6900, Heidelberg, Federal Republic of Germany

    T. Eisele & R. Lang (Heisenberg fellow)

Authors
  1. T. Eisele
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Lang
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported by Deutsche Forschungsgemeinschaft

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Eisele, T., Lang, R. Asymptotics for the wiener sausage with drift. Probab. Th. Rel. Fields 74, 125–140 (1987). https://doi.org/10.1007/BF01845643

Download citation

  • Received: 21 October 1985

  • Accepted: 21 July 1986

  • Issue Date: March 1987

  • DOI: https://doi.org/10.1007/BF01845643

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

  • Computer Simulation
  • Lower Bound
  • Stochastic Process
  • Brownian Motion
  • Decay Rate
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