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Generalized displacement convexity for nonlinear mobility continuity equation and entropy power concavity on Wasserstein space over Riemannian manifolds

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Abstract

In this paper, we prove the generalized displacement convexity for nonlinear mobility continuity equation with p-Laplacian on Wasserstein space over Riemannian manifolds under the generalized McCann condition GMC(m, n). Moreover, we obtain some variational formulae along the Langevin deformation of flows on the generalized Wasserstein space, which is the interpolation between the gradient flow and the geodesic flow. We also establish the connection between the displacement convexity of entropy functionals and the concavity of p-Rényi entropy powers. As an application, we derive the NIW formula which indicates the relationship between the p-Rényi entropy powers \({\mathcal {N}}_{b}\), the Fisher information \({\mathcal {I}}_{b}\) and the \({\mathcal {W}}\)-entropy.

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Acknowledgements

The first author would like to thank Professor Xiang-Dong Li for his interest and valuable discussion, particularly on the entropy power and NIW formula. The authors are thankful to the anonymous reviewers and editors for their constructive comments and suggestions on the earlier version for this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, China

    Yu-Zhao Wang, Sheng-Jie Li & Xinxin Zhang

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  1. Yu-Zhao Wang
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  2. Sheng-Jie Li
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  3. Xinxin Zhang
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Correspondence to Yu-Zhao Wang.

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Y. Wang: Research of Y.-Z. Wang has been supported by NSFC No. 11701347.

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Wang, YZ., Li, SJ. & Zhang, X. Generalized displacement convexity for nonlinear mobility continuity equation and entropy power concavity on Wasserstein space over Riemannian manifolds. manuscripta math. 172, 405–426 (2023). https://doi.org/10.1007/s00229-022-01415-w

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  • DOI: https://doi.org/10.1007/s00229-022-01415-w

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