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 Dello Schiavo \cite{schiavo2020rademacher}, which establishes a version of Rademacher's theorem
on Wasserstein spaces.
This is joint work with Pedram Emami.