The Annals of Probability

Large deviations for random walk in a space–time product environment

Atilla Yilmaz

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Abstract

We consider random walk (Xn)n≥0 on ℤd in a space–time product environment ω∈Ω. We take the point of view of the particle and focus on the environment Markov chain (Tn, Xnω)n≥0 where T denotes the shift on Ω. Conditioned on the particle having asymptotic mean velocity equal to any given ξ, we show that the empirical process of the environment Markov chain converges to a stationary process μξ under the averaged measure. When d≥3 and ξ is sufficiently close to the typical velocity, we prove that averaged and quenched large deviations are equivalent and when conditioned on the particle having asymptotic mean velocity ξ, the empirical process of the environment Markov chain converges to μξ under the quenched measure as well. In this case, we show that μξ is a stationary Markov process whose kernel is obtained from the original kernel by a Doob h-transform.

Article information

Source
Ann. Probab., Volume 37, Number 1 (2009), 189-205.

Dates
First available in Project Euclid: 17 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1234881688

Digital Object Identifier
doi:10.1214/08-AOP400

Mathematical Reviews number (MathSciNet)
MR2489163

Zentralblatt MATH identifier
1159.60355

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60F10: Large deviations

Keywords
Dynamical random environment rare events Doob h-transform

Citation

Yilmaz, Atilla. Large deviations for random walk in a space–time product environment. Ann. Probab. 37 (2009), no. 1, 189--205. doi:10.1214/08-AOP400. https://projecteuclid.org/euclid.aop/1234881688


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