Keynote Speaker:Xie Longjie

Abstract:

We study the asymptotic behavior for an inhomogeneous multiscale stochastic dynamical system with singular 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. This is based on a joint work with Michael Roeckner.

Introduction to the Speaker:

Xie Longjie, professor of Jiangsu Normal University, received his bachelor's degree from Central South University in 2011 and his doctorate from Wuhan University in 2016. He was a visiting scholar at the University of Illinois at Urbana Champaign (UIUC) from 2017 to 2018, and a Humboldt scholar at the Bielefeld University in Germany from 2018 to 2020, and won the 14th Zhong Jiaqing Mathematical Award in 2019. His main research directions include thermonuclear estimation of nonlocal operators and stochastic (partial) differential equations with singular coefficients. Up to now, his 20 papers have been published in well-known journals such asProbab Theory Rel Fields,Ann Probab,Ann Inst Henri Poincare-Pr,Stoch Proc ApplandJ Diff Equ.

Inviter:

Chen Zhang, Professor of School of Mathematics

Time:

13:30-14:30 on December 3 (Thursday)

Location:

Tencent Meeting, ID: 706435159

Sponsored by: School of Mathematics, Shandong University