{
  "id": "0708.2101",
  "version": "v4",
  "published": "2007-08-15T20:41:26.000Z",
  "updated": "2008-02-25T15:44:58.000Z",
  "title": "Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time",
  "authors": [
    "Julien Randon-Furling",
    "Satya N. Majumdar"
  ],
  "comment": "13 pages, 5 figures. Published in Journal of Statistical Mechanics: Theory and Experiment (J. Stat. Mech. (2007) P10008, doi:10.1088/1742-5468/2007/10/P10008)",
  "journal": "Journal of Statistical Mechanics: Theory and Experiment (2007) P10008",
  "doi": "10.1088/1742-5468/2007/10/P10008",
  "categories": [
    "cond-mat.stat-mech",
    "math.PR"
  ],
  "abstract": "We calculate analytically the probability density $P(t_m)$ of the time $t_m$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\\sim t_m^{-3/2}$ for large $t_m$ and $P(t_m)\\sim t_m^{-1/2}$ for small $t_m$. In presence of a drift towards the origin, $P(t_m)$ decays exponentially for large $t_m$. The results from numerical simulations are in excellent agreement with our analytical predictions.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2008-02-25T15:44:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first-passage time",
      "distribution",
      "continuous-time brownian motion",
      "joint probability density",
      "excellent agreement"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2007,
      "month": "Oct",
      "volume": 2007,
      "number": 10,
      "pages": 10008
    },
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007JSMTE..10....8R"
    }
  }
}