Speaker : Prof. Xiangdong Li (CAS)
Title : Generalized Dyson Brownian motion, McKean-Vlasov equation and fluctuation of eigenvalus of random matrices
Time : 2013-12-02 (Mon) 15:00 -
Place : Seminar Room 617, Institute of Mathematics (NTU Campus)
Abstract: Dyson Brownian motion is an interacting -particle system with the logarithmic Coulomb interaction and has been used in various areas in mathematics and physics. In this talk, we present some recent results in the study of the generalized Dyson Brownian motion (GDBM) and the associated McKean-Vlasov equation with general external potential . Under suitable condition on , we prove the existence and uniqueness of the strong solution to SDE for GDBM, and prove that the large limit of any weak convergent subsequence satisfies the non-linear McKean-Vlasov equation. Using Otto's infinite dimensional Riemannian geometry on the Wasserstein space, we prove that, if for a constant , then the McKean-Vlasov equation for GDBM has a unique weak solution. This yields the Law of the Large Numbers for GDBM. We also prove the longtime convergence of the McKean-Vlasov equation for convex . In a work in progress, we study the fluctuation of GDBM and prove the Central Limit Theorem for the empirical measure of GDBM. Some open problems will be discussed. This is a joint work with my PhD students Songzi Li (Fudan U. and Toulouse U.) and Yongxiao Xie (AMSS, CAS).