On large deviations of coupled diffusions with time scale separation

Abstract: We consider two Ito equations that evolve on different time scales. The equations are fully coupled in the sense that all coefficients may depend on both the "slow" and the "fast" processes and the diffusion terms may be correlated. The diffusion term in the "slow" process is small. A large deviation principle is obtained for the joint distribution of the slow process and of the empirical measure of the fast process. By projecting on the slow and fast variables, we arrive at new results on large deviations in the averaging framework and on large deviations of the empirical measures of ergodic diffusions, respectively. The proof of the main result relies on the property that exponential tightness implies large deviation relative compactness. The identification of the large deviation rate function is accomplished by analysing the large deviation limit of an exponential martingale.