Large deviations for a scalar diffusion in random environment

Abstract

Let $σ(u)$, $u\in \mathbb{R}$ be an ergodic stationary Markov chain, taking a finite number of values $a_1,...,a_m$, and $b(u)=g(σ(u))$, where $g$ is a bounded and measurable function. We consider the diffusion type process $$ dX^ε_t = b(X^ε_t/ε)dt + ε^κσ\big(X^ε_t/ε\big)dB_t, t\le T $$ subject to $X^ε_0=x_0$, where $ε$ 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 $κ<1/6$, the family $\{X^ε_t\}_{ε\to 0}$ satisfies the Large Deviations Principle (LDP) of the Freidlin-Wentzell type with the constant drift $\mathbf{b}$ and the diffusion $\mathbf{a}$, given by $$ \mathbf{b}=\sum\limits_{i=1}^m\dfrac{g(a_i)}{a^2_i}π_i\Big/ \sum\limits_{i=1}^m\dfrac{1}{a^2_i}π_i, \quad \mathbf{a}=1\Big/\sum\limits_{i=1}^m\dfrac{1}{a^2_i}π_i, $$ where $\{π_1,...,π_m\}$ is the invariant distribution of the chain $σ(u)$.