{"ID":2876082,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.01768","arxiv_id":"2509.01768","title":"Totally convex functions, $L^2$-Optimal transport for laws of random measures, and solution to the Monge problem","abstract":"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 $L^2(\\mathrm{Q},\\mathbb{M};\\mathrm{H}).$","short_abstract":"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 $\\\\ma...","url_abs":"https://arxiv.org/abs/2509.01768","url_pdf":"https://arxiv.org/pdf/2509.01768v1","authors":"[\"Alessandro Pinzi\",\"Giuseppe Savaré\"]","published":"2025-09-01T21:03:21Z","proceeding":"math.FA","tasks":"[\"math.FA\",\"math.OC\",\"math.PR\"]","methods":"[]","has_code":false}