Optimal maps in essentially non-branching spaces

TL;DR

This paper proves that in essentially non-branching metric measure spaces satisfying the measure contraction property, optimal transport plans are induced by unique maps when the initial measure is absolutely continuous.

Contribution

It establishes the existence and uniqueness of optimal transport maps in a broad class of metric measure spaces under specific curvature and non-branching conditions.

Findings

Optimal plans are induced by maps in these spaces.

Uniqueness of the transport plan is guaranteed.

Results extend the theory of optimal transport in non-smooth settings.

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.