Averaging Principle and Normal Deviations for Multiscale Stochastic Systems

We study the asymptotic behavior for an inhomogeneous multiscale stochastic dynamical system with non-smooth coefficients. Depending on averaging regime and homogenization regime, two strong convergences in principle of functional law large numbers type are established. Then we consider small fluctuations around its average. Nine cases central limit theorems obtained. In particular, even though averaged equation original is same, corresponding normal deviation can be quite different due to difference interactions between fast scales scales. provide intuitive explanations each case. Furthermore, sharp rates both obtained, these shown rely only regularity coefficients respect slow variable, do not depend their which coincide intuition since equations component has been totally or homogenized out.