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.
"Large deviations for random walk in a space–time product environment." Ann. Probab. 37 (1) 189 - 205, January 2009. https://doi.org/10.1214/08-AOP400