Journal of Statistical Physics

, Volume 41, Issue 5–6, pp 811–824 | Cite as

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

  • M. Khantha
  • V. Balakrishnan


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.

Key words

Continuous time random walk biased random walk reflection principle escape time distribution 


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Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • M. Khantha
    • 1
  • V. Balakrishnan
    • 1
  1. 1.Department of PhysicsIndian Institute of TechnologyMadrasIndia

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