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arXiv · 1609.00782

Optimal maps in essentially non-branching spaces

Fabio Cavalletti·Andrea Mondino
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Abstract

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.

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Fabio Cavalletti, Andrea Mondino. 2016-09-03. Optimal maps in essentially non-branching spaces. https://doi.org/10.1142/s0219199717500079

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