On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion

doi:10.1007/s10955-020-02627-8

	On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion
	Li, Songzi 3; Li, Xiang-Dong 1,4; Xie, Yong-Xiao 2
刊名	JOURNAL OF STATISTICAL PHYSICS
	2020-09-21
页码	29
关键词	Generalized Dyson Brownian motion McKean-Vlasov equation Gradient flow Optimal transportation Voiculescu free entropy Law of Large Numbers Propagation of chaos
ISSN号	0022-4715
DOI	10.1007/s10955-020-02627-8
英文摘要	We study the generalized Dyson Brownian motion (GDBM) of an interacting N-particle system with logarithmic Coulomb interaction and general potential V. Under reasonable condition on V, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures ofGDBMis tight on C([0, T], P(R)) and all the large N limits satisfy a nonlinear McKean-Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, and Blower, we prove that the McKean-Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space of probability measures over R. Using the optimal transportation theory, we prove that if V '' >= K for some constant K is an element of R, the McKean-Vlasov equation has a unique weak solution in the space of probability measures P(R). This establishes the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM with non-quadratic external potentials which are not necessarily convex. Finally, we prove the longtime convergence of the McKean-Vlasov equation for C-2-convex potentials V.
资助项目	NSFC[11601287] ; NSFC[11771430] ; NSFC[11901569] ; Key Laboratory RCSDS, CAS[2008DP173182]
WOS研究方向	Physics
语种	英语
出版者	SPRINGER
WOS记录号	WOS:000571672100001
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/52205]
专题	中国科学院数学与系统科学研究院
通讯作者	Li, Xiang-Dong
作者单位	1.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China 2.Shandong Normal Univ, Sch Math & Stat, 1 Univ Rd,Sci & Technol Pk, Jinan 250385, Peoples R China 3.Renmin Univ China, Sch Math, 59 Zhongguancun Da Jie, Beijing 100872, Peoples R China 4.Chinese Acad Sci, Acad Math & Syst Sci, 55 Zhongguancun East Rd, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Li, Songzi,Li, Xiang-Dong,Xie, Yong-Xiao. On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion[J]. JOURNAL OF STATISTICAL PHYSICS,2020:29.
APA	Li, Songzi,Li, Xiang-Dong,&Xie, Yong-Xiao.(2020).On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion.JOURNAL OF STATISTICAL PHYSICS,29.
MLA	Li, Songzi,et al."On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion".JOURNAL OF STATISTICAL PHYSICS (2020):29.