Fast-slow stochastic dynamical system with singular coefficients  

作　　者：Longjie Xie 

机构地区：[1]School of Mathematics and Statistics,Jiangsu Normal University,Xuzhou 221116,China

出　　处：《Science China Mathematics》2023年第4期819-838,共20页中国科学：数学（英文版）

基　　金：supported by the Alexander von Humboldt foundation and National Natural Science Foundation of China(Grant Nos.12090011,12071186 and 11931004)。

摘　　要：This paper aims to study the asymptotic behavior of a fast-slow stochastic dynamical system with singular coefficients,where the fast motion is given by a continuous diffusion process while the slow component is driven by anα-stable noise withα∈[1,2).Using Zvonkin’s transformation and the technique of the Poisson equation,we have that both the strong and weak convergences in the averaging principle are established,which can be viewed as a functional law of large numbers.Then we study the small fluctuations between the original system around its average.We show that the normalized difference converges weakly to an Ornstein-Uhlenbeck type Gaussian process,which is a form of the functional central limit theorem.Furthermore,sharp rates for the above convergences are also obtained,and these convergences are shown to not depend on the regularities of the coefficients with respect to the fast variable,which reflect the effects of noises on the multi-scale systems.

关 键 词：multi-scale dynamical system averaging principle central limit theorem α-stable process Zvonkin’s transformation 

分 类 号：O19[理学—数学]