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.

我们研究了具有非光滑系数的非均匀多尺度随机动力系统的渐近行为。依赖于平均机制和均质化机制，在大数类型的函数定律的平均原理上建立了两个强大的收敛性。然后，我们考虑系统围绕其平均值的微小波动。获得了9个泛函中心极限类型定理的案例。特别是，即使原始系统的平均方程是相同的，由于快速标度和偏差标度之间相互作用的差异，法向偏差的相应均化极限可能会非常不同。对于每种情况，我们都提供了非常直观的解释。此外，