{"paper":{"title":"Monotone Measure-Preserving Maps in Hilbert Spaces: Existence, Uniqueness, and Stability","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Alberto Gonz\\'alez-Sanz, Bodhisattva Sen, Marc Hallin","submitted_at":"2023-05-19T15:39:51Z","abstract_excerpt":"The contribution of this work is twofold. The first part deals with a Hilbert-space version of McCann's celebrated result on the existence and uniqueness of monotone measure-preserving maps: given two probability measures $\\rm P$ and $\\rm Q$ on a separable Hilbert space $\\mathcal{H}$ where $\\rm P$ does not give mass to \"small sets\" (namely, Lipschitz hypersurfaces), we show, without imposing any moment assumptions, that there exists a gradient of convex function $\\nabla\\psi$ pushing ${\\rm P} $ forward to ${\\rm Q}$. In case $\\mathcal{H}$ is infinite-dimensional, ${\\rm P}$-a.s. uniqueness is not"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.11751","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.11751/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}