{
  "id": "2002.00859",
  "version": "v1",
  "published": "2020-02-03T16:12:16.000Z",
  "updated": "2020-02-03T16:12:16.000Z",
  "title": "Isometric study of Wasserstein spaces --- the real line",
  "authors": [
    "György Pál Gehér",
    "Tamás Titkos",
    "Dániel Virosztek"
  ],
  "comment": "32 pages, 7 figures. Accepted for publication in Trans. Amer. Math. Soc",
  "categories": [
    "math.MG",
    "math-ph",
    "math.FA",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $\\mathcal{W}_2\\left(\\mathbb{R}^n\\right)$. It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute $\\mathrm{Isom}\\left(\\mathcal{W}_p(\\mathbb{R})\\right)$, the isometry group of the Wasserstein space $\\mathcal{W}_p(\\mathbb{R})$ for all $p \\in [1, \\infty)\\setminus\\{2\\}$. We show that $\\mathcal{W}_2(\\mathbb{R})$ is also exceptional regarding the parameter $p$: $\\mathcal{W}_p(\\mathbb{R})$ is isometrically rigid if and only if $p\\neq 2$. Regarding the underlying space, we prove that the exceptionality of $p=2$ disappears if we replace $\\mathbb{R}$ by the compact interval $[0,1]$. Surprisingly, in that case, $\\mathcal{W}_p\\left([0,1]\\right)$ is isometrically rigid if and only if $p\\neq1$. Moreover, $\\mathcal{W}_1\\left([0,1]\\right)$ admits isometries that split mass, and $\\mathrm{Isom}\\left(\\mathcal{W}_1\\left([0,1]\\right)\\right)$ cannot be embedded into $\\mathrm{Isom}\\left(\\mathcal{W}_1(\\mathbb{R})\\right).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-03T16:12:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54E40",
      "46E27",
      "60A10",
      "60B05"
    ],
    "keywords": [
      "real line",
      "isometric study",
      "isometry group",
      "exotic isometry flow",
      "quadratic wasserstein space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}