The Monge-Kantorovich problem on Wasserstein space

We consider the Monge-Kantorovich problem between two random measuress. More precisely, given probability measures mathbbP1,mathbbP2∈P(P(M)) on the space P(M) of probability measures on a smooth compact manifold, we study the optimal transport problem between mathbbP1 and mathbbP2 where the cost function is given by the squared Wasserstein distance W22(μ,nu) between μ,nu ∈ P(M). Under appropriate assumptions on mathbbP1, 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(W2(μ,nu)), for strictly convex and strictly increasing functions h, is also established. The proofs rely heavily on a recent result of Schiavo citeschiavo2020rademacher, which establishes a version of Rademacher's theorem on Wasserstein space.