{
  "id": "1811.02417",
  "version": "v1",
  "published": "2018-11-06T15:30:51.000Z",
  "updated": "2018-11-06T15:30:51.000Z",
  "title": "Universality for persistence exponents of local times of self-similar processes with stationary increments",
  "authors": [
    "Christian Mönch"
  ],
  "comment": "23 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We show that $\\mathbb{P} ( \\ell_X(0,T] \\leq 1)=(c_X+o(1))T^{-(1-H)}$, where $\\ell_X$ is the local time measure at $0$ of any recurrent $H$-self-similar real-valued process $X$ with stationary increments that admits a sufficiently regular local time and $c_X$ is some constant depending only on $X$. A special case is the Gaussian setting, i.e. when the underlying process is fractional Brownian motion, in which our result settles a conjecture by Molchan [Commun. Math. Phys. 205, 97-111 (1999)] who obtained the upper bound $1-H$ on the decay exponent of $\\mathbb{P} ( \\ell_X(0,T] \\leq 1)$. Our approach establishes a new connection between persistence probabilities and Palm theory for self-similar random measures, thereby providing a general framework which extends far beyond the Gaussian case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-06T15:30:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G22",
      "60G15",
      "60G18"
    ],
    "keywords": [
      "stationary increments",
      "self-similar processes",
      "persistence exponents",
      "universality",
      "sufficiently regular local time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}