{"ID":2833470,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.03726","arxiv_id":"2512.03726","title":"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.","short_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....","url_abs":"https://arxiv.org/abs/2512.03726","url_pdf":"https://arxiv.org/pdf/2512.03726v1","authors":"[\"Christophe Vauthier\"]","published":"2025-12-03T12:14:49Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}