Large deviations for a scalar diffusion in random environment

Let $sigma(u)$, $uin mathbb{R}$ 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 $$ dX^epsilon_t = b(X^epsilon_t/epsilon)dt + epsilon^kappasigmabig(X^epsilon_t/epsilonbig)dB_t, tle T $$ 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 ${X^epsilon_t}_{epsilonto 0}$ satisfies the Large Deviations Principle (LDP) of the Freidlin-Wentzell type with the constant drift $mathbf{b}$ and the diffusion $mathbf{a}$, given by $$ mathbf{b}=sumlimits_{i=1}^mdfrac{g(a_i)}{a^2_i}pi_iBig/ sumlimits_{i=1}^mdfrac{1}{a^2_i}pi_i, quad mathbf{a}=1Big/sumlimits_{i=1}^mdfrac{1}{a^2_i}pi_i, $$ where ${pi_1,...,pi_m}$ is the invariant distribution of the chain $sigma(u)$.