Large deviations for two scaled diffusions

Abstract

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 $(X^{\epsilon },{\nu }^{\epsilon })$ with ${\nu }^{\epsilon }(dt,dz)$ being an occupation type measure corresponding to ${\xi }_t^{\epsilon }$. In some sense we obtain a combination of Freidlin-Wentzell's and Donsker-Varadhan's results. Our approach relies the concept of the exponential tightness and Puhalskii's theorem.