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$\boldsymbol~W$-entropy formulas on super Ricci flowsand Langevin deformation on Wasserstein spaceover Riemannian manifolds

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  • ReceivedJul 26, 2017
  • AcceptedDec 16, 2017
  • PublishedMay 31, 2018

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 onsuper-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula forthe Ricci flow, we provethe $W$-entropy formula for the heat equation associated with the Witten Laplacian on $n$-dimensional complete Riemannian manifolds with the $CD(K,~m)$-condition, andthe $W$-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 $K\in~\mathbb{R}$ and $m\in~[n,~\infty]$. Furthermore, we provean 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) onthe displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity betweenabove two $W$-entropy formulas, we introduce the Langevin deformation of geometric flows onthe 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.


Acknowledgment

The first author was supported by a Postdoctoral Fellowship at Beijing Normal University and China Postdoctoral Science Foundation (Grant No. 2017M610797). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11771430 and 11371351) and Key Laboratory of Random Complex Structures and Data Science, Chinese Academy of Sciences (Grant No. 2008DP173182). The authors thank Professors Shigeki Aida, Dominique Bakry, Jean-Michel Bismut, Jean-Dominique Deuschel, David Elworthy, Kazuhiro Kuwae, Michel Ledoux, Ngaiming Mok, Karl-Theodor Sturm, Anton Thalmaier, Fengyu Wang and Dr. Yuzhao Wang for their interests, comments and helpful discussions during various stages of this work. Moreover, the authors thank an anonymous referee for his very careful reading and very nice comments which led them to improve the paper. The authors thank the Editors for their interests on this work and for their nice comment which suggests the authors to include some further problems and comments for future work and possible progress on the topic discussed in this paper. In 2016, the second author of this paper was invited to give a Special Invited Talk in the 2016 Autumn Meeting of the Mathematical Society of Japan. This survey is an improved version of the abstract for this meeting. The second author thanks the committee members of the Mathematical Society of Japan, in particular, Professors Shigeki Aida and Kazuhiro Kuwae, for their interests and invitation. Finally, the authors thank the Mathematical Society of Japan to allow them to submit this survey to SCIENCE CHINA Mathematics for publication.


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