Existence and uniqueness of optimal transport maps in locally compact $CAT(0)$ spaces

TL;DR

This paper proves the existence and uniqueness of optimal transport maps in certain $CAT(0)$ spaces, including Riemannian manifolds of non-positive curvature, and introduces a polar factorization theorem in these spaces.

Contribution

It establishes conditions for unique optimal transport maps in $CAT(0)$ spaces and extends polar factorization results to this setting.

Findings

Unique optimal transport maps exist in certain $CAT(0)$ spaces.

The results apply to Riemannian manifolds with non-positive curvature.

A polar factorization theorem is developed for $CAT(0)$ spaces.

Abstract

We show that in a locally compact complete $CAT(0)$ space satisfying positive angles property and a disintegration regularity for its canonical Hausdorff measure, there exists a unique optimal transport map that push-forwards a given absolutely continuous probability measure to another probability measure. In particular this holds for the Riemannian manifolds of non-positive sectional curvature and $CAT(0)$ Euclidean polyhedral complexes. Moveover we give a polar factorization result for Borel maps in $CAT(0)$ spaces in terms of optimal transport maps and measure preserving maps.