Large deviations for a scalar diffusion in random environment

Let $\sigma(u)$, $u\in \mathbb{R}$ be an ergodic stationary Markov chain, taking a finite number of values $a_1,...,a_m$, and $b(u)=g(\sigma(u))$, where $g$ is a bounded and measurable function. We consider the diffusion type process $$dX^\epsilon_t = b(X^\epsilon_t/\epsilon)dt + \epsilon^\kappa\sigma\big(X^\epsilon_t/\epsilon\big)dB_t, t\le T$$ subject to $X^\epsilon_0=x_0$, where $\epsilon$ is a small positive parameter, $B_t$ is a Brownian motion, independent of $\sigma$, and $\kappa> 0$ is a fixed constant. We show that for $\kappa<1/6$, the family $\{X^\epsilon_t\}_{\epsilon\to 0}$ satisfies the Large Deviations Principle (LDP) of the Freidlin-Wentzell type with the constant drift $\mathbf{b}$ and the diffusion $\mathbf{a}$, given by $$\mathbf{b}=\sum\limits_{i=1}^m\dfrac{g(a_i)}{a^2_i}\pi_i\Big/ \sum\limits_{i=1}^m\dfrac{1}{a^2_i}\pi_i, \quad \mathbf{a}=1\Big/\sum\limits_{i=1}^m\dfrac{1}{a^2_i}\pi_i,$$ where $\{\pi_1,...,\pi_m\}$ is the invariant distribution of the chain $\sigma(u)$.