{
  "id": "2109.03192",
  "version": "v1",
  "published": "2021-09-07T16:56:21.000Z",
  "updated": "2021-09-07T16:56:21.000Z",
  "title": "Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry",
  "authors": [
    "Lorenzo Dello Schiavo",
    "Kohei Suzuki"
  ],
  "comment": "82 pages, 3 diagrams, 2 tables",
  "categories": [
    "math.PR",
    "math-ph",
    "math.FA",
    "math.MG",
    "math.MP"
  ],
  "abstract": "We construct a canonical differential structure on the configuration space $\\boldsymbol\\Upsilon$ over a singular base space $X$ and with a general invariant measure $\\mu$ on $\\boldsymbol\\Upsilon$. We present an analytic structure on $\\boldsymbol\\Upsilon$, constructing a strongly local Dirichlet form $\\mathcal E$ on $L^2(\\boldsymbol\\Upsilon, \\mu)$ for $\\mu$ in a large class of probability measures. We then investigate the geometric structure of the extended metric measure space $\\boldsymbol\\Upsilon$ endowed with the $L^2$-transportation extended distance $\\mathsf{d}_{\\boldsymbol\\Upsilon}$ and with the measure $\\mu$. By establishing Rademacher- and Sobolev-to-Lipschitz-type properties for $\\mathcal E$, we finally provide a complete identification of the analytic and the geometric structure -- the canonical differential structure induced on $\\boldsymbol\\Upsilon$ by $X$ and $\\mu$ -- showing that $\\mathcal E$ coincides with the Cheeger energy of $(\\boldsymbol\\Upsilon,\\mathsf{d}_{\\boldsymbol\\Upsilon},\\mu)$ and that the intrinsic distance of $\\mathcal E$ coincides with $\\mathsf{d}_{\\boldsymbol\\Upsilon}$. The class of base spaces to which our results apply includes sub-Riemannian manifolds, RCD spaces, and path/loop spaces over Riemannian manifolds; as for $\\mu$ our results include quasi-Gibbs measures, in particular: Poisson measures, canonical Gibbs measures, as well as some determinantal/permanental point processes (sine$_\\beta$, Airy$_\\beta$, Bessel$_{\\alpha,\\beta}$, Ginibre). A number of applications to interacting particle systems and infinite-dimensional metric measure geometry are also discussed. In particular, we prove the universality of the $L^2$-transportation distance $\\mathsf{d}_{\\boldsymbol\\Upsilon}$ for the Varadhan short-time asymptotics for diffusions on $\\boldsymbol\\Upsilon$, regardless of the choice of $\\mu$. Many of our results are new even in the case of Euclidean spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-07T16:56:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C25",
      "30L99",
      "70F45",
      "60G55",
      "60G57"
    ],
    "keywords": [
      "configuration space",
      "singular spaces",
      "canonical differential structure",
      "dirichlet-form",
      "base space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 82,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}