{
  "id": "1602.06770",
  "version": "v1",
  "published": "2016-02-22T13:41:26.000Z",
  "updated": "2016-02-22T13:41:26.000Z",
  "title": "Temporal correlations of the running maximum of a Brownian trajectory",
  "authors": [
    "O. Benichou",
    "P. L. Krapivsky",
    "C. Mejia-Monasterio",
    "G. Oshanin"
  ],
  "comment": "5 pages, 4 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math.PR"
  ],
  "abstract": "We study the correlations between the maxima $m$ and $M$ of a Brownian motion on the time intervals $[0,t_1]$ and $[0,t_2]$, with $t_2>t_1$. We determine exact forms of the distribution functions $P(m,M)$ and $P(G = M - m)$, and calculate the moments $\\mathbb{E}\\{\\left(M - m\\right)^k\\}$ and the cross-moments $\\mathbb{E}\\{m^l M^k\\}$ with arbitrary integers $l$ and $k$. We compute the Pearson correlation coefficient $\\rho(m,M)$ and show that $\\rho(m,M) \\sim \\sqrt{t_1/t_2}$ when $t_2 \\to \\infty$ with $t_1$ kept fixed, revealing strong memory effects in the statistics of the maxima of a Brownian motion. As an application, we discuss a possibility of extracting the ensemble-average diffusion coefficient in single-trajectory experiments using a single realization of the maximum process of a Brownian motion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-22T13:41:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian trajectory",
      "temporal correlations",
      "running maximum",
      "brownian motion",
      "ensemble-average diffusion coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}