Moderate deviations in the averaging principle of a SDE with small diffusion in a random environment (Q2738893)

Consider an SDE in \({\mathbb R}^d\): NEWLINE\[NEWLINEdX^\varepsilon_t=b(X^\varepsilon_t,\xi_{t/\varepsilon}) dt +\sqrt{\varepsilon}a(X^\varepsilon_t,\xi_{t/\varepsilon}) dW_t, \quad X_0^\varepsilon=x_0, \qquad t\in[0,1],NEWLINE\]NEWLINE where \(\xi\) is a Markov process (``random environment'') independent of the Wiener process \(W\). Let \(\bar{b}(x)=\int b(x,z)\pi(dz)\), where \(\pi\) is an invariant distribution for \(\xi\), and let \(\bar x\) be the solution of the deterministic system \(\dot{x}_t={\bar b}({\bar x}_t)\), \({\bar x}_0=x\). An asymptotic behaviour, as \(\varepsilon\searrow 0\), of the process \(\eta_t^\varepsilon={X_t^\varepsilon-{\bar x}_t\over \sqrt{\varepsilon}h(\varepsilon)}\) in the space \(C_0([0,1],{\mathbb R}^d)\) is investigated. Under some ergodicity assumptions on \(\xi\) and regularity conditions on \(a,b\), it is proved that \(\eta^\varepsilon\) satisfies the moderate deviation principle (``moderate'' means that \(h\) is assumed to satisfy \(\lim_{\varepsilon\to 0}h(\varepsilon)=\infty\) and \({\sqrt\varepsilon} h(\varepsilon)=0\) rather than \(h(\varepsilon)=1/\sqrt\varepsilon\), as it is in the classical large deviation case). The corresponding rate function is given explicitly.