W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

doi:10.1007/s11425-017-9227-7

CORC > 数学与系统科学研究院 > 中国科学院数学与系统科学研究院 > 应用数学研究所

	W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds
	Li, Songzi 1; Li, Xiang-Dong2,3
刊名	SCIENCE CHINA-MATHEMATICS
	2018-08-01
卷号	61 期号:8 页码:1385-1406
关键词	W-entropy Witten Laplacian Langevin deformation (K, m)-super Ricci flows
ISSN号	1674-7283
DOI	10.1007/s11425-017-9227-7
英文摘要	In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K,m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a (K,m)-super Ricci flow, where K a R and m a [n,a]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result improves an important result due to Lott and Villani (2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.
资助项目	Postdoctoral Fellowship at Beijing Normal University ; China Postdoctoral Science Foundation[2017M610797] ; National Natural Science Foundation of China[11771430] ; National Natural Science Foundation of China[11371351] ; Key Laboratory of Random Complex Structures and Data Science, Chinese Academy of Sciences[2008DP173182]
WOS研究方向	Mathematics
语种	英语
出版者	SCIENCE PRESS
WOS记录号	WOS:000440140500002
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/30851]
专题	应用数学研究所
通讯作者	Li, Xiang-Dong
作者单位	1.Beijing Normal Univ, Sch Math Sci, Beijing 100875, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China 3.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
推荐引用方式 GB/T 7714	Li, Songzi,Li, Xiang-Dong. W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds[J]. SCIENCE CHINA-MATHEMATICS,2018,61(8):1385-1406.
APA	Li, Songzi,&Li, Xiang-Dong.(2018).W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds.SCIENCE CHINA-MATHEMATICS,61(8),1385-1406.
MLA	Li, Songzi,et al."W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds".SCIENCE CHINA-MATHEMATICS 61.8(2018):1385-1406.