Totally convex functions, $L^2$-Optimal transport for laws of random measures, and solution to the Monge problem
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arXiv
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| Natura: | Preprint |
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2025
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| author | Pinzi, Alessandro Savaré, Giuseppe |
| author_facet | Pinzi, Alessandro Savaré, Giuseppe |
| contents | 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 $L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Totally convex functions, $L^2$-Optimal transport for laws of random measures, and solution to the Monge problem Pinzi, Alessandro Savaré, Giuseppe Functional Analysis Optimization and Control Probability Primary: 49Q22 (Primary), 46N30, 60G15, 28A33, 60B05 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 $L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$ |
| title | Totally convex functions, $L^2$-Optimal transport for laws of random measures, and solution to the Monge problem |
| topic | Functional Analysis Optimization and Control Probability Primary: 49Q22 (Primary), 46N30, 60G15, 28A33, 60B05 |
| url | https://arxiv.org/abs/2509.01768 |