Non-branching geodesics and optimal maps in strong CD(K,\infty )-spaces |
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Authors: | Tapio Rajala Karl-Theodor Sturm |
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Institution: | 1. University of Jyv?skyl?, Department of Mathematics and Statistics, P.O. Box 35 (MaD), 40014, University of Jyv?skyl?, Finland 2. Institut für Angewandte Mathematik, Universit?t Bonn, Endenicher Allee 60, 53115, Bonn, Germany
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Abstract: | We prove that in metric measure spaces where the entropy functional is \(K\) -convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map. The results are applicable in metric measure spaces having Riemannian Ricci curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below by some constant. |
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