演講公告

新聞標題：　( 2013-12-03 )

• 演講主題：Generalized Dyson Brownian Motion, McKean-Vlasov Equation and Fluctuation of Eigenvalues of Random Matrices

• 主講人：李向東教授(中國科學院)

• 演講日期：2013年12月10日(星期二) 下午2:00 –3:00

• 演講地點：(光復校區) 科學一館223室

• 茶會時間：當天下午1:30 (科學一館205室)

• 摘要內容：
  

  Dyson Brownian motion is an interacting $N$-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 the generalized Dyson Brownian motion (GDBM) and the associated McKean-Vlasov equation with general external potential $V$. Under suitable condition on $V$, we prove the existence and uniqueness of the strong solution to SDE for GDBM, and prove that the large $N$ limit of any weak convergent subsequence of the empirical measure of GDBM satisfies the non-linear McKean-Vlasov equation. Using Otto's infinite dimensional Riemannian geometry on the Wasserstein space, we prove that, if $V''\geq K$ for a constant $K\in \mathbb{R}$, 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 long time convergence of the McKean-Vlasov equation for convex $V$. 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).

• 相關檔案：Talk_1021210.doc

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