Existence and uniqueness of optimal transport maps

TL;DR

This paper establishes the existence and uniqueness of optimal transport maps in certain metric measure spaces under a new weak measure property, extending previous results and showing stability under measure changes.

Contribution

It introduces a new weak property ensuring optimal transport map existence and uniqueness, covering spaces beyond those with measure contraction property.

Findings

Existence and uniqueness of optimal transport maps under the new property.

Stability of the property under measure changes with positive continuous functions.

The new property is weaker than the measure contraction property.

Abstract

Let $(X,d,m)$ be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided $(X,d,m)$ satisfies a new weak property concerning the behavior of $m$ under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property. We also prove a stability property for Assumption 1: If $(X,d,m)$ satisfies Assumption 1 and $\tilde m = g\cdot m$, for some continuous function $g >0$, then also $(X,d,\tilde m)$ verifies Assumption 1. Since these changes in the reference measures do not preserve any Ricci type curvature bounds, this shows that our condition is strictly weaker than measure contraction property.