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Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates

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Abstract

The duality in Bakry–Émery’s gradient estimates and Wasserstein controls for heat distributions is extended to that in refined estimates in a high generality. As a result, we find an equivalent condition to Bakry–Ledoux’s refined gradient estimate involving an upper dimension bound. This new condition is described as a \(L^2\)-Wasserstein control for heat distributions at different times. The \(L^p\)-version of those estimates are studied on Riemannian manifolds via coupling method.

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Acknowledgments

The author would like to tell his gratitude to the anonymous referee. His/Her comments help the author to improve the quality of the paper. Especially, Proposition 4.4 and Remark 4.6 are essentially due to the comment. This work is partially supported by the Japan Society for the Promotion of Science Grant-in-Aid for Young Scientists (B) 22740083.

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  1. Graduate School of Science, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo, 152-8551, Japan

    Kazumasa Kuwada

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  1. Kazumasa Kuwada
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Correspondence to Kazumasa Kuwada.

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Communicated by L. Ambrosio.

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Kuwada, K. Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates. Calc. Var. 54, 127–161 (2015). https://doi.org/10.1007/s00526-014-0781-2

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