{
  "id": "1909.05594",
  "version": "v1",
  "published": "2019-09-12T12:19:14.000Z",
  "updated": "2019-09-12T12:19:14.000Z",
  "title": "Time between the maximum and the minimum of a stochastic process",
  "authors": [
    "Francesco Mori",
    "Satya N. Majumdar",
    "Gregory Schehr"
  ],
  "comment": "Main text: 5 pages + 3 Figs, Supp. Mat.: 20 pages + 7 Figs",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We present an exact solution for the probability density function $P(\\tau=t_{\\min}-t_{\\max}|T)$ of the time-difference between the minimum and the maximum of a one-dimensional Brownian motion of duration $T$. We then generalise our results to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. We demonstrate that these results can be directly applied to study the position-difference between the minimal and the maximal height of a fluctuating $(1+1)$-dimensional Kardar-Parisi-Zhang interface on a substrate of size $L$, in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for L\\'evy flights and find that it differs from the Brownian motion result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-12T12:19:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic process",
      "brownian motion result",
      "probability density function",
      "periodic brownian motion",
      "finite jump variance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}