Moderate Deviation Principle for dynamical systems with small random perturbation

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.