{
  "id": "2301.00262",
  "version": "v1",
  "published": "2022-12-31T18:14:43.000Z",
  "updated": "2022-12-31T18:14:43.000Z",
  "title": "Curvature bound of Dyson Brownian Motion",
  "authors": [
    "Kohei Suzuki"
  ],
  "comment": "33 pages, comments welcome!",
  "categories": [
    "math.PR",
    "math-ph",
    "math.DG",
    "math.FA",
    "math.MP"
  ],
  "abstract": "In this article, we show $1$-Bakry-\\'Emery lower Ricci curvature bound $\\mathrm{BE}_1(0, \\infty)$ of a Dirichlet form on the configuration space whose invariant measure is $\\mathsf{sine}_\\beta$ ensemble for any $\\beta>0$. As a particular case of $\\beta=2$, our result proves $\\mathrm{BE}_1(0, \\infty)$ for a Dirichlet form related to the unlablled Dyson Brownian motion. We prove furthermore several functional inequalities including the integral Bochner inequality, the local Poincar\\'e and the local log-Sobolev inequalities as well as the log-Harnack and the dimension-free Harnack inequalities, the Lipschitz contraction property and the $L^\\infty$-to-Lipschitz regularisation property of the semigroup with the $L^2$-transportation-type extended distance. At the end of the article, we provide a sufficient condition for the synthetic lower Ricci curvature bound in the case of general invariant measures beyond~$\\mathsf{sine}_\\beta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-31T18:14:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dyson brownian motion",
      "bakry-emery lower ricci curvature bound",
      "synthetic lower ricci curvature bound",
      "dirichlet form",
      "inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}