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w-entropy formulas on super Ricci flows and Langevin deformation on wasserstein space over Riemannian manifolds Dedicated to Professor Zhiming Ma on His Seventieth Birthday

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Science China Mathematics 2018年第8期61卷 1385-1406页

作者： Songzi Li Xiang-Dong Li School of Mathematical Sciences Beijing Normal University Beijing 100875 China Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China

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 ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the w-entropy formula for the geodesic flow on the wasserstein space over Riemannian *** 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.

关键词： w-entropy witten Laplacian Langevin deformation (K,m)-super Ricci flows

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