Non-branching geodesics and optimal maps in strong CD(K,∞) -spaces
Rajala, T., & Sturm, K.-T. (2014). Non-branching geodesics and optimal maps in strong CD(K,∞) -spaces. Calculus of Variations and Partial Differential Equations, 50(3-4), 831-846. https://doi.org/10.1007/s00526-013-0657-x
© Springer-Verlag Berlin Heidelberg 2013. This is a final draft version of an article whose final and definitive form has been published by Springer. Published in this repository with the kind permission of the publisher.
We prove that in metric measure spaces where the entropy functional is Kconvex 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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