The Monge-Kantorovich problem on 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 $W_2^2(\hat{I}¼,\hat{I}½)$ between $\hat{I}¼,\hat{I}½\in \mathcal{P}(M)$. 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 $h(W_2(\hat{I}¼,\hat{I}½))$, 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.