Moderate deviations for a diffusion-type process in a random environment

Let σ(u), u ∈, be an ergodic stationary Markov chain, taking a finite number of values a1, ⋯ , a_m, and let b(u) = g(σ(u)), where g is a bounded and measurable function. We consider the diffusion-type process (Mathematic equation present) subject to (Mathematic equation present), where e is a small positive parameter, B_t is a Brownian motion, independent of σ, and κ < 0 is a fixed constant. We show that for κ > (Mathematic equation present), the family (Mathematic equation present) satisfies the large deviation principle (LDP) of Freidlin-Wentzell type with the constant drift b and the diffusion a, given by (Mathematic equation present) where {p1, ⋯ , p_m} is the invariant distribution of the chain σ(u).