KMS Of Academy of mathematics and systems sciences, CAS
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 Publication | SCIENCE CHINA-MATHEMATICS |
ISSN | 1674-7283 |
Volume | 61Issue:8Pages:1385-1406 |
Abstract | 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. |
Keyword | W-entropy Witten Laplacian Langevin deformation (K, m)-super Ricci flows |
DOI | 10.1007/s11425-017-9227-7 |
Language | 英语 |
Funding Project | 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 Research Area | Mathematics |
WOS Subject | Mathematics, Applied ; Mathematics |
WOS ID | WOS:000440140500002 |
Publisher | SCIENCE PRESS |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/30852 |
Collection | 应用数学研究所 |
Affiliation | 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 |
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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