{
  "id": "cond-mat/0306101",
  "version": "v1",
  "published": "2003-06-04T17:39:54.000Z",
  "updated": "2003-06-04T17:39:54.000Z",
  "title": "Persistence Exponents and the Statistics of Crossings and Occupation Times for Gaussian Stationary Processes",
  "authors": [
    "George M. C. A. Ehrhardt",
    "Satya N. Majumdar",
    "Alan J. Bray"
  ],
  "comment": "35 pages, 31 figures",
  "journal": "Phys. Rev. E 69, 016106 (2004)",
  "doi": "10.1103/PhysRevE.69.016106",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We consider the persistence probability, the occupation-time distribution and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian Stationary Processes (GSPs) of zero mean. We first consider the Ornstein-Uhlenbeck process, finding expressions for the mean and variance of the number of crossings and the `partial survival' probability. We then elaborate on the correlator expansion developed in an earlier paper [G. C. M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2001)] to calculate discretely sampled persistence exponents of GSPs of known correlator by means of a series expansion in the correlator. We apply this method to the processes d^n x/dt^n=\\eta(t) with n > 2, incorporating an extrapolation of the series to the limit of continuous sampling. We extend the correlator method to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings. We apply these general methods to the d^n x/dt^n=\\eta(t) processes for n=1 (random walk), n=2 (random acceleration) and larger n, and to diffusion from random initial conditions in 1-3 dimensions. The results for discrete sampling are extrapolated to the continuum limit where possible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-06-04T17:39:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "occupation time",
      "persistence exponents",
      "distribution",
      "statistics",
      "correlator"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Phys. Rev. E"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}