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Lagrangian calculus for nonsymmetric diffusion operators

  • Christian Ketterer EMAIL logo

Abstract

We characterize lower bounds for the Bakry–Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy and line integrals on the L2-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable 1-form in the sense of [N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 2018, 1196, 1–161]. This extends the Lott–Sturm–Villani approach for lower Ricci curvature bounds of metric measure spaces. In generalized smooth context, consequences are new Bishop–Gromov estimates, pre-compactness under measured Gromov–Hausdorff convergence, and a Bonnet–Myers theorem that generalizes previous results by Kuwada [K. Kuwada, A probabilistic approach to the maximal diameter theorem, Math. Nachr. 286 2013, 4, 374–378]. We show that N-warped products together with lifted vector fields satisfy the curvature-dimension condition. For smooth Riemannian manifolds, we derive an evolution variational inequality and contraction estimates for the dual semigroup of nonsymmetric diffusion operators. Another theorem of Kuwada [K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal. 258 2010, 11, 3758–3774], [K. Kuwada, Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates, Calc. Var. Partial Differential Equations 54 2015, 1, 127–161] yields Bakry–Emery gradient estimates.

MSC 2010: 58J05; 53C21

Communicated by Frank Duzaar


Acknowledgements

This work was done while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2016 semester, supported by the National Science Foundation under Grant No. DMS-1440140. The author thanks the organizers of the Differential Geometry Program and MSRI for providing a great environment for research. The author is also very grateful to the unknown reviewer for his careful reading of the article and his remarks.

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Received: 2018-01-12
Revised: 2018-03-27
Accepted: 2018-04-10
Published Online: 2018-04-20
Published in Print: 2020-10-01

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