The Annals of Probability

Random walks in dynamic random environments: A transference principle

Frank Redig and Florian Völlering

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We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties for the environment as seen from the position of the walker, that is, the environment process. We can transfer the rate of mixing in time of the environment to the rate of mixing of the environment process with a loss of at most polynomial order. Therefore the method is applicable to environments with sufficiently fast polynomial mixing. We obtain unique ergodicity of the environment process. Moreover, the unique invariant measure of the environment process depends continuously on the jump rates of the walker.

As a consequence we obtain the law of large numbers and a central limit theorem with nondegenerate variance for the position of the walk.

Article information

Ann. Probab., Volume 41, Number 5 (2013), 3157-3180.

First available in Project Euclid: 12 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]
Secondary: 60F17: Functional limit theorems; invariance principles

Environment process coupling random walk transference principle central limit theorem


Redig, Frank; Völlering, Florian. Random walks in dynamic random environments: A transference principle. Ann. Probab. 41 (2013), no. 5, 3157--3180. doi:10.1214/12-AOP819.

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