Large deviations for two scaled diffusions

We formulate large deviations principle (LDP) for diffusion pair $(X^epsilon,xi^epsilon)=(X_t^epsilon,xi_t^epsilon)$, where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for $(X^epsilon, u^epsilon)$ with $ u^epsilon(dt,dz)$ being an occupation type measure corresponding to $xi_t^epsilon$. In some sense we obtain a combination of Freidlin-Wentzells and Donsker-Varadhans results. Our approach relies the concept of the exponential tightness and Puhalskiis theorem.