Moderate Deviation Principle for dynamical systems with small random perturbation

Abstract: Consider the stochastic differential equation in $\rr^d$ dX^{\e}_t&=b(X^{\e}_t)dt+\sqrt{\e}\sigma(X^\e_t)dB_t X^{\e}_0&=x_0,\quad x_0\in\rr^d$ where $b:\rr^d\to\rr^d$ is $C^1$ such that $<x,b(x)> \leq C(1+|x|^2)$, $\sigma:\rr^d\to \MM(d\times n)$ is locally Lipschitzian with linear growth, and $B_t$ is a standard Brownian motion taking values in $\rr^n$. Freidlin-Wentzell's theorem gives the large deviation principle for $X^\e$ for small $\e$. In this paper we establish its moderate deviation principle.