Optimal transport with optimal transport cost: the Monge-Kantorovich problem on Wasserstein spaces

We consider the Monge-Kantorovich problem between two random measures. More precisely, given probability measures P1,P2 is an element of P(P(M))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}_1,{\mathbb {P}}_2\in {\mathcal {P}}({\mathcal {P}}(M))$$\end{document} on the space P(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}(M)$$\end{document} of probability measures on a smooth compact manifold, we study the optimal transport problem between P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}_1$$\end{document} and P2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}_2 $$\end{document} where the cost function is given by the squared Wasserstein distance W22(mu,nu)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_2<^>2(\mu ,\nu )$$\end{document} between mu,nu is an element of P(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu ,\nu \in {\mathcal {P}}(M)$$\end{document}. Under appropriate assumptions on P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}_1$$\end{document}, 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(mu,nu))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(W_2(\mu ,\nu ))$$\end{document}, 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.