Optimal maps in essentially non-branching spaces

By Fabio Cavalletti, Andrea Mondino. 2016

66 citations recorded by the source corpus.

In this note we prove that in a metric measure space $(X, d, m)$ verifying the measure contraction property with parameters $K \in \mathbb{R}$ and $1< N< \infty$, any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to $m$ and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/82c9d12a9a3ee4648f1e781d8b036b8e8e9b9b63

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The globalization theorem for the Curvature-Dimension condition (2016, background): “ established in [14] on non-branching spaces (see also the work of Erbar–Kuwada–Sturm [35, Corollary 3.13, Theorem 3.14 and Remark 3.26] for an extension to the essentially non-branching setting, cf. [67, 29]). Let us also mention here the work of Ambrosio–Mondino–Savar´e [10], who independently of ”