Variational Analysis in the Wasserstein Hierarchy

Abstract

Let $M$ be a complete connected Riemannian manifold. For $n \geq 0$, we endow the Wasserstein space $P^{(n)}_2(M) = P_2(\ldots P_2(M)\ldots)$, equipped with the Wasserstein distance $W_2$, with a variational structure that generalizes the standard variational structure on $P_2(M)$ provided by optimal transport theory. Our approach makes use of tools from category theory to lift the geometric structure of the manifold $M$ to the spaces $P^{(n)}_2(M)$, in order to establish in a principled way a rigorous theoretical framework for variational analysis on the space $P^{(n)}_2(M)$. In particular, we obtain a precise characterization of the constant speed geodesics of the space $P^{(n)}_2(M)$ in terms of optimal velocity plans. Moreover, we introduce a notion of gradient for functionals defined on $P^{(n)}_2(M)$, which allows us to study the differentiability and the convexity of various types of such functionals.