Totally convex functions, $L^2$-Optimal transport for laws of random measures, and solution to the Monge problem

We study the Optimal Transport problem for laws of random measures in the Kantorovich-Wasserstein space $\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$, associated with a Hilbert space $\mathrm{H}$ (with finite or infinite dimension) and for the corresponding quadratic cost induced by the squared Wasserstein metric in $\\mathcal{P}_2(\mathrm{H}).$ Despite the lack of smoothness of the cost, the fact that the space $\mathcal{P}_2(\mathrm{H})$ is not Hilbertian, and the curvature distortion induced by the underlying Wasserstein metric, we will show how to recover at the level of random measures in $\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$ the same deep and powerful results linking Euclidean Optimal Transport problems in $\mathcal{P}_2(\mathrm{H})$ and convex analysis. Our approach relies on the notion of totally convex functionals, on their total subdifferentials, and their Lagrangian liftings in the space square integrable $\mathrm{H}$-valued maps $L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$ With these tools, we identify a natural class of regular measures in $\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$ for which the Monge formulation of the OT problem has a unique solution and we will show that this class includes relevant examples of measures with full support in $\mathcal{P}_2(\mathrm{H})$ arising from the push-forward transformation of nondegenerate Gaussian measures in $L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$