We consider the Monge–Kantorovich problem between two random measures. 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 (J Funct Anal 278(6):108397, 2020), which establishes a version of Rademacher’s theorem on Wasserstein spaces.
