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