Monotonicity and rigidity of the \({\mathcal {W}}\)-entropy on \({\mathsf {RCD}} (0,N)\) spaces

Abstract

By means of a space-time Wasserstein control, we show the monotonicity of the \({\mathcal {W}}\)-entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipation of the \({\mathcal {W}}\)-entropy is also proved. These extend known results even on weighted Riemannian manifolds in some respects. In addition, we reveal that some singular spaces will exhibit the rigidity models while only the Euclidean space does in the class of smooth weighted Riemannian manifolds.

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Acknowledgements

This work was started when the authors attended the Workshop on Geometry and Probability held at the Luxembourg University in October 2013. The authors would like to thank Anton Thalmaier for his invitations which lead this work possible. The first author warmly thanks to Shouhei Honda for his suggestion to formulate Theorem 5.8 and to Nicola Gigli for an improvement of Theorem 5.8. He also wish to tell his gratitude to Karl-Theodor Sturm for fruitful discussions. Especially, Theorem 5.6 comes from a discussion with him. The first author was supported by the European Union through the ERC–AdG “Ricci Bounds” for Prof. K. T. Sturm. The second author would like to thank Songzi Li for fruitful collaboration on the study of the W-entropy for heat flows, geodesic and Langevin flows on the Wasserstein space over manifolds. He also would like to express his gratitude to Kazuhiro Kuwae and Yu–Zhao Wang for valuable discussions on various topics related to this work.

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Correspondence to Xiang-Dong Li.

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K. Kuwada: Supported in part by JSPS Grant-in-Aid for Young Scientist (A) (KAKENHI) 26707004.

X.-D. Li: Research supported by NSFC No. 11771430, Key Laboratory RCSDS, CAS, No. 2008DP173182, and Hua Luo-Keng Research Grant of AMSS, CAS (2015-2017).

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Kuwada, K., Li, XD. Monotonicity and rigidity of the \({\mathcal {W}}\)-entropy on \({\mathsf {RCD}} (0,N)\) spaces. manuscripta math. 164, 119–149 (2021). https://doi.org/10.1007/s00229-019-01177-y

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Mathematics Subject Classification (2010)

  • Primary 53C23
  • 53C44
  • 53J35
  • Secondary 51f99
  • 58J65
  • 60J60