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

doi:10.1007/s10955-020-02627-8

CSpace > 应用数学研究所

	On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion
	Li, Songzi 1; Li, Xiang-Dong2,3 ; Xie, Yong-Xiao 4
	2020-09-21
发表期刊	JOURNAL OF STATISTICAL PHYSICS
ISSN	0022-4715
页码	29
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.
关键词	Generalized Dyson Brownian motion McKean-Vlasov equation Gradient flow Optimal transportation Voiculescu free entropy Law of Large Numbers Propagation of chaos
DOI	10.1007/s10955-020-02627-8
收录类别	SCI
语种	英语
资助项目	NSFC[11601287] ; NSFC[11771430] ; NSFC[11901569] ; Key Laboratory RCSDS, CAS[2008DP173182]
WOS研究方向	Physics
WOS类目	Physics, Mathematical
WOS记录号	WOS:000571672100001
出版者	SPRINGER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/52206
专题	应用数学研究所
通讯作者	Li, Xiang-Dong
作者单位	1.Renmin Univ China, Sch Math, 59 Zhongguancun Da Jie, Beijing 100872, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, 55 Zhongguancun East Rd, Beijing 100190, Peoples R China 3.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China 4.Shandong Normal Univ, Sch Math & Stat, 1 Univ Rd,Sci & Technol Pk, Jinan 250385, 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.