The Monge-Kantorovich problem on Wasserstein space

We consider the Monge-Kantorovich problem between two random measuress. More precisely, given probability measures P₁, P₂ (P (M) ) on the space P (M) of probability measures on a smooth compact manifold, we study the optimal transport problem between P₁ and P₂ where the cost function is given by the squared Wasserstein distance W₂² (, ) between, P (M). Under appropriate assumptions on P₁, 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 (, ) ), for strictly convex and strictly increasing functions h, is also established. The proofs rely heavily on a recent result of Schiavo schiavo2020rademacher, which establishes a version of Rademacher's theorem on Wasserstein space.