The Monge-Kantorovich problem on Wasserstein space

TL;DR

This paper investigates the Monge-Kantorovich optimal transport problem between probability measures on the Wasserstein space over a compact manifold, establishing existence, uniqueness, and structure of optimal plans under certain conditions.

Contribution

It proves the existence and uniqueness of optimal transport plans as maps for measures on Wasserstein space, extending results to general cost functions of the form h(W_2).

Findings

Existence of a unique optimal plan as a map under certain conditions

Extension of results to cost functions h(W_2) with convex h

Relies on Rademacher's theorem in Wasserstein space

Abstract

We consider the Monge-Kantorovich problem between two random measuress. More precisely, given probability measures $\mathbb{P}_1,\mathbb{P}_2\in\mathcal{P}(\mathcal{P}(M))$ on the space $\mathcal{P}(M)$ of probability measures on a smooth compact manifold, we study the optimal transport problem between $\mathbb{P}_1$ and $\mathbb{P}_2 $ where the cost function is given by the squared Wasserstein distance $W_2^2(\mu,\nu)$ between $\mu,\nu \in \mathcal{P}(M)$. Under appropriate assumptions on $\mathbb{P}_1$, we prove that there exists a unique optimal plan and that it takes the form of an optimal map. An extension of this result to cost functions of the form $h(W_2(\mu,\nu))$, for strictly convex and strictly increasing functions $h$, is also established. The proofs rely heavily on a recent result of Schiavo \cite{schiavo2020rademacher}, which establishes a version of Rademacher's theorem on Wasserstein space.