Journal of Statistical Physics

, Volume 84, Issue 3–4, pp 415–454 | Cite as

A Brownian motion version of the directed polymer problem

  • Joseph G. Conlon
  • Peder A. Olsen


Consider a Brownian particle in three dimensions in a random environment. The environment is determined by a potential random in space and time. It is shown that at small noise the large-time behavior of the particle is diffusive. The diffusion constant depends on the environment. This work generalizes previous results for random walk in a random environment. In these results the diffusion constant does not depend on the environment.

Key Words

Directed polymer Brownian motion transfer matrix spectral gap 


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  1. 1.
    E. Bolthausen, A note on the diffusion of directed polymers in a random environment,Commun. Math. Phys. 123:529–534 (1989).Google Scholar
  2. 2.
    L. Breiman,Probability (Addison-Wesley, Reading, Massachusetts, 1968).Google Scholar
  3. 3.
    J. Z. Imbrie and T. Spencer, Diffusion of directed polymers in a random environment,J. Stat. Phys. 52:609–626 (1988).Google Scholar
  4. 4.
    M. Krasnoselskii,Positive Solutions of Operator Equations (Noordhoff, Groningen, 1964).Google Scholar
  5. 5.
    D. Ruelle,Statistical Mechanics-Rigorous Results (Benjamin, 1969).Google Scholar
  6. 6.
    A. Taylor,Functional Analysis (Wiley, New York, 1958).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Joseph G. Conlon
    • 1
  • Peder A. Olsen
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn Arbor

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