Skip to main content
Log in

Reflection principles for biased random walks and application to escape time distributions

Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We present a reflection principle for an arbitrarybiased continuous time random walk (comprising both Markovian and non-Markovian processes) in the presence of areflecting barrier on semi-infinite and finite chains. For biased walks in the presence of a reflecting barrier this principle (which cannot be derived from combinatorics) is completely different from its familiar form in the presence of an absorbing barrier. The result enables us to obtain closed-form solutions for the Laplace transform of the conditional probability for biased walks on finite chains for all three combinations of absorbing and reflecting barriers at the two ends. An important application of these solutions is the calculation of various first-passage-time and escape-time distributions. We obtain exact results for the characteristic functions of various kinds of escape time distributions for biased random walks on finite chains. For processes governed by a long-tailed event-time distribution we show that the mean time of escape from bounded regions diverges even in the presence of a bias—suggesting, in a sense, the absence of true long-range diffusion in such “frozen” processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. S. Chandrasekhar,Rev. Mod. Phys. 15:1 (1943).

    Google Scholar 

  2. W. Feller,An Introduction to Probability Theory and Its Applications, Vols. 1 and 2 (Wiley, New York, 1966).

    Google Scholar 

  3. E. W. Montroll and B. J. West, inFluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979).

    Google Scholar 

  4. V. Balakrishnan and M. Khantha,Pramana 21:187 (1983).

    Google Scholar 

  5. D. R. Cox,Renewal Theory (Methuen, London, 1967).

    Google Scholar 

  6. V. Balakrishnan, inStochastic Processes—Formalism and Applications, G. S. Agarwal and S. Dattagupta, eds. (Lecture Notes in Physics, No. 184, Springer, Berlin, 1983).

    Google Scholar 

  7. V. Balakrishnan and G. Venkataraman,Pramana 16:109 (1981).

    Google Scholar 

  8. M. Lax and H. Scher,Phys. Rev. Lett. 39:781 (1977).

    Google Scholar 

  9. M. Khantha and V. Balakrishnan,Phys. Rev. B 29:4679 (1984).

    Google Scholar 

  10. K. W. Kehr and J. W. Haus,Physica A93:412 (1978).

    Google Scholar 

  11. V. Balakrishnan,Pramana 17:55 (1981).

    Google Scholar 

  12. M. Khantha and V. Balakrishnan,J. Phys. C 16:6291 (1983).

    Google Scholar 

  13. M. Khantha and V. Balakrishnan,Pramana 21:111 (1983).

    Google Scholar 

  14. A. J. F. Siegert,Phys. Rev. 81:617 (1951).

    Google Scholar 

  15. D. A. Darling and A. J. F. Siegert,Ann. Math. Stat. 24:624 (1953).

    Google Scholar 

  16. E. W. Montroll and G. H. Weiss,J. Math. Phys. 6:167 (1965).

    Google Scholar 

  17. E. W. Montroll,J. Math. Phys. 10:753 (1969).

    Google Scholar 

  18. V. Seshadri and B. J. West,Proc. Natl. Acad. Sci. (USA) 79:4501 (1982).

    Google Scholar 

  19. M. F. Shlesinger,J. Stat. Phys. 10:421 (1974).

    Google Scholar 

  20. H. Scher and M. Lax,Phys. Rev. B 7:4491 (1973).

    Google Scholar 

  21. H. Scher and E. W. Montroll,Phys. Rev. B 12:2455 (1975).

    Google Scholar 

  22. J. K. E. Tunaley,J. Stat. Phys. 24:587 (1976).

    Google Scholar 

  23. M. F. Shlesinger and B. D. Hughes,Physica A109:597 (1981).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khantha, M., Balakrishnan, V. Reflection principles for biased random walks and application to escape time distributions. J Stat Phys 41, 811–824 (1985). https://doi.org/10.1007/BF01010005

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01010005

Key words

Navigation