Large deviations for a scalar diffusion in random environment

Abstract

Let $\ddot{I}ƒ(u)$, $u\in \mathbb{R}$ be an ergodic stationary Markov chain, taking a finite number of values $a_1,...,a_m$, and $b(u)=g(\ddot{I}ƒ(u))$, where $g$ is a bounded and measurable function. We consider the diffusion type process $$d{X}^{\hat{I}}{\mu }_t=b(X^{\hat{I}}{\mu }_t/\hat{I}\mu )dt+\hat{I}{\mu }^{\hat{I}}º\ddot{I}ƒ(X^{\hat{I}}{\mu }_t/\hat{I}\mu )d{B}_t,t\le T$$ subject to $X^{\hat{I}}{\mu }_0=x_0$, where $\hat{I}\mu$ is a small positive parameter, $B_t$ is a Brownian motion, independent of $\ddot{I}ƒ$, and $\hat{I}º>0$ is a fixed constant. We show that for $\hat{I}º<1/6$, the family $\{X^{\hat{I}}{\mu }_t{\}}_{\hat{I}\mu \to 0}$ satisfies the Large Deviations Principle (LDP) of the Freidlin-Wentzell type with the constant drift $\mathbf{b}$ and the diffusion $\mathbf{a}$, given by $$\mathbf{b}=\sum _{i=1}^m\frac{g(a_i)}{a_i^2}\ddot{I}{€}_i/\sum _{i=1}^m\frac{1}{a_i^2}\ddot{I}{€}_i,\mathbf{a}=1/\sum _{i=1}^m\frac{1}{a_i^2}\ddot{I}{€}_i,$$ where $\{\ddot{I}{€}_1,...,\ddot{I}{€}_m\}$ is the invariant distribution of the chain $\ddot{I}ƒ(u)$.

15 pages