Mass Transportation on Sub-Riemannian Manifolds

TL;DR

This paper extends optimal transport theory to sub-Riemannian manifolds, proving existence, uniqueness, and regularity of optimal maps, with specific results in the Heisenberg group.

Contribution

It generalizes Brenier-McCann's Theorem to sub-Riemannian settings, establishing regularity and differentiability properties of optimal transport maps.

Findings

Existence and uniqueness of optimal transport maps in sub-Riemannian manifolds.

Absolute continuity of Wasserstein geodesics.

Approximate differentiability of optimal maps in the Heisenberg group.

Abstract

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the Monge-Amp\`ere equation.