Averaging principle of SDE with small diffusion: Moderate deviations

Abstract

Consider the following stochastic differential equation in $\mathbb{R}^d$: \begin{eqnarray*} dX^\varepsilon_t &=& b(X_t^\varepsilon,\xi_{t/\varepsilon})\,dt +\sqrt{\varepsilon} a(X_t^\varepsilon,\xi_{t/\varepsilon})\,dW_t,\\ X^\varepsilon_0 &=& x_0, \end{eqnarray*} where the random environment $(\xi_t)$ is an exponentially ergodic Markov process, independent of the Wiener process $(W_t)$, with invariant probability measure $\pi$, and $\varepsilon$ is some small parameter. In this paper we prove the moderate deviations for the averaging principle of $X^\varepsilon$, that is, deviations of $(X^\varepsilon_t)$ around its limit averaging system $(\bar x_t)$ given by %$\bar{d}$. % $d\bar x_t=\bar b(\bar x_t)\,dt$ and $\bar x_0=x_0$ where $\bar b(x)=\mathbb{E}_\pi(b(x,\cdot))$. More precisely we obtain the large deviation estimation about \[ \Bigl(\eta^\varepsilon_t={X^\varepsilon_t-\bar x_t \over \sqrt{\varepsilon} h(\varepsilon)}\Big)_{t\in[0,1]} \] in the space of continuous trajectories, as $\varepsilon$ decreases to 0, where $h(\varepsilon)$ is some deviation scale satisfying $1\ll h(\varepsilon)\ll\varepsilon^{-1/2}$. Our strategy will be first to show the exponential tightness and then the local moderate deviation principle, which relies on some new method involving a conditional Schilder's theorem and a moderate deviation principle for inhomogeneous integral functionals of Markov processes, previously established by the author.