Large Deviations for Random Walk in a Space-time Product Environment

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 μ ξ 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 μ ξ under the quenched measure as well. In this case, we show that μ ξ is a stationary Markov process whose kernel is obtained from the original kernel by a Doob h-transform.