{"paper":{"title":"Wasserstein Diffusion on Multidimensional Spaces","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.FA","math.MG"],"primary_cat":"math.PR","authors_text":"Karl-Theodor Sturm","submitted_at":"2024-01-23T12:46:02Z","abstract_excerpt":"Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space ${\\mathcal P}(M)$ of probability measures on $M$ that is\n  (i) reversible w.r.t.~the entropic measure ${\\mathbb P}^\\beta$ on ${\\mathcal P}(M)$, heuristically given as $$d\\mathbb{P}^\\beta(\\mu)=\\frac{1}{Z} e^{-\\beta \\, \\text{Ent}(\\mu| m)}\\ d\\mathbb{P}^*(\\mu);$$ (ii) associated with a regular Dirichlet form with carr\\'e du champ derived from the Wasserstein gradient in the sense of Otto calculus $${\\mathcal E}_W(f)=\\liminf_{g\\to f}\\ \\frac12\\int_{{\\mathcal P}(M)} \\big\\|\\nabla_W g\\big\\|^2(\\mu)\\ d{\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.12721","kind":"arxiv","version":3},"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/2401.12721/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"}