Non-branching geodesics and optimal maps in strong CD(K,{\infty})-spaces

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.

15 pages, 3 figures

References in corpus (1)

• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure

Cited by in corpus (3)

• On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces
• Failure of topological rigidity results for the measure contraction property
• Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth Have Finitely Generated Fundamental Groups