Averaging principle and normal deviations for multi-scale stochastic systems
A talk in the Bielefeld Stochastic Afternoon series by
Longjie Xie from JSNU
| Abstract: | We study the asymptotic behavior for a in-homogeneous multi-scale stochastic system with singular coefficients. Depending on the averaging regime and the homogenization regime, two strong convergences of functional law of large numbers type are obtained. Then we consider the small fluctuations of the singularly perturbed diffusion around its averaged motion. Nine cases of functional central limit type theorem are established. In particular, even though the averaged equation for the original system is the same, the corresponding homogenization limit for the normal derivations can be quite different due to the difference of interactions between the fast scales and the deviation scales. We provide quite intuitive explanations for each cases. The method to prove our main results is rather simple and unified. Furthermore, sharp rates both for the strong convergences and the functional central limit theorems are also obtained, and these convergences are shown to rely only on the regularities of the coefficients of the system with respect to the slow variable, and do not depend on their regularities with respect to the fast variable. |