{
  "id": "2211.15609",
  "version": "v1",
  "published": "2022-11-28T18:08:32.000Z",
  "updated": "2022-11-28T18:08:32.000Z",
  "title": "Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity",
  "authors": [
    "Nina Holden",
    "Yizheng Yuan"
  ],
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the SLE with its natural parametrization. More precisely, denoting by $d\\in(0,2]$ the dimension of the curve, we show the following. 1. The optimal $\\psi$-variation is $\\psi(x)=x^d(\\log\\log x^{-1})^{-(d-1)}$ in the sense that $\\eta$ is of finite $\\psi$-variation for this $\\psi$ and not for any function decaying more slowly as $x \\downarrow 0$. 2. The optimal modulus of continuity is $\\omega(s) = c\\,s^{1/d}(\\log s^{-1})^{1-1/d}$, i.e. $|\\eta(t)-\\eta(s)| \\le \\omega(t-s)$ for this $\\omega$, and not for any function decaying more slowly as $s \\downarrow 0$. 3. $\\limsup_{t\\downarrow 0} |\\eta(t)|\\,(t^{1/d}(\\log\\log t^{-1})^{1-1/d})^{-1}$ is a deterministic constant in $(0,\\infty)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-28T18:08:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J67",
      "60G17",
      "60G18",
      "30C20"
    ],
    "keywords": [
      "schramm-loewner evolution",
      "up-to-constant variation",
      "continuity",
      "optimal modulus",
      "function decaying"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}