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Einstein relation for reversible random walks in random environment on Z

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  • Lam, Hoang-Chuong
  • Depauw, Jerome

Abstract

The aim of this paper is to consider reversible random walk in a random environment in one dimension and prove the Einstein relation for this model. It says that the derivative at 0 of the effective velocity under an additional local drift equals the diffusivity of the model without drift (Theorem 1.2). Our method here is very simple: we solve the Poisson equation (Pω−I)g=f and then use the pointwise ergodic theorem in Wiener (1939) [10] to treat the limit of the solutions to obtain the desired result. There are analogous results for Markov processes with discrete space and for diffusions in random environment.

Suggested Citation

  • Lam, Hoang-Chuong & Depauw, Jerome, 2016. "Einstein relation for reversible random walks in random environment on Z," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 983-996.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:4:p:983-996
    DOI: 10.1016/j.spa.2015.10.007
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    References listed on IDEAS

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    1. Lebowitz, Joel L. & Rost, Hermann, 1994. "The Einstein relation for the displacement of a test particle in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 54(2), pages 183-196, December.
    2. Komorowski, T. & Olla, S., 2005. "Einstein relation for random walks in random environments," Stochastic Processes and their Applications, Elsevier, vol. 115(8), pages 1279-1301, August.
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