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W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds
Li, Songzi1; Li, Xiang-Dong2,3
2018-08-01
Source PublicationSCIENCE CHINA-MATHEMATICS
ISSN1674-7283
Volume61Issue:8Pages:1385-1406
AbstractIn 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.
KeywordW-entropy Witten Laplacian Langevin deformation (K, m)-super Ricci flows
DOI10.1007/s11425-017-9227-7
Language英语
Funding ProjectPostdoctoral 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 Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000440140500002
PublisherSCIENCE PRESS
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/30852
Collection应用数学研究所
Affiliation1.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
Recommended Citation
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.
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