Large Deviations for a Scalar Diffusion in Random Environment

Let ξ(u), u ∈ R be an ergodic stationary Markov chain, taking a finite number of values, and consider the diffusion process generated by the SDE dX t = b(X ε t )dt+ ε ξ ( X t /ε ) dBt with a small positive scaling parameter ε, where B = (Bt)t∈R+ is a Brownian motion, independent of ξ, and κ ≥ 0 is a fixed constant. Such model describes evolution of a particle, perturbed by a small white noise disturbance, whose intensity is switched by the random environment ξ. We show that for κ ∈ (0, 1/6), the process X satisfies the same Large Deviations Principle (LDP) of the Freidlin-Wentzell type as the process X̂: dX t = b(X̂ ε t )dt+ ε κ √ adBt, with a = 1 Eξ−2(0) . For κ = 0, X converges weakly to the the solution of the SDE dXt = b(Xt)dt+ √ adBt.