Non-branching geodesics and optimal maps in strong $CD(K,\infty)$-spaces
Calc. Var. Partial Differential Equations
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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