{
  "id": "1507.06238",
  "version": "v1",
  "published": "2015-07-22T16:10:14.000Z",
  "updated": "2015-07-22T16:10:14.000Z",
  "title": "The Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory",
  "authors": [
    "Mathieu Delorme",
    "Kay Joerg Wiese"
  ],
  "comment": "5 pages, 7 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "Fractional Brownian motion is a non-Markovian Gaussian process $X_t$, indexed by the Hurst exponent $H$. It generalises standard Brownian motion (corresponding to $H=1/2$). We study the probability distribution of the maximum $m$ of the process and the time $t_{\\rm max}$ at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting $H=1/2 + \\varepsilon$. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of $H$ test these analytical predictions and show excellent agreement, even for large $\\varepsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-22T16:10:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional brownian motion",
      "analytic results",
      "perturbation theory",
      "generalises standard brownian motion",
      "non-markovian gaussian process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}