{"paper":{"title":"Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MG","math.MP"],"primary_cat":"math.PR","authors_text":"Kohei Suzuki, Lorenzo Dello Schiavo","submitted_at":"2021-09-07T16:56:21Z","abstract_excerpt":"We construct a canonical differential structure on the configuration space $\\Upsilon$ over a singular base space $X$ and with a general invariant measure $\\mu$ on $\\Upsilon$. We present an analytic structure on $\\Upsilon$, constructing a strongly local Dirichlet form $\\mathcal E$ on $L^2(\\Upsilon, \\mu)$ for $\\mu$ in a large class of probability measures. We then investigate the geometric structure of the extended metric measure space $\\Upsilon$ endowed with the $L^2$-transportation extended distance $\\mathsf{d}_{\\Upsilon}$ and with the measure $\\mu$. By establishing Rademacher- and Sobolev-to-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.03192","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/2109.03192/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"}