{
  "id": "2105.11042",
  "version": "v1",
  "published": "2021-05-23T23:24:59.000Z",
  "updated": "2021-05-23T23:24:59.000Z",
  "title": "Markovian structure in the concave majorant of Brownian motion",
  "authors": [
    "Mehdi Ouaki",
    "Jim Pitman"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "The purpose of this paper is to highlight some hidden Markovian structure of the concave majorant of the Brownian motion. Several distributional identities are implied by the joint law of a standard one-dimensional Brownian motion $B$ and its almost surely unique concave majorant $K$ on $[0,\\infty)$. In particular, the one-dimensional distribution of $2 K_t - B_t$ is that of $R_5(t)$, where $R_5$ is a $5-$dimensional Bessel process with $R_5(0) = 0$. The process $2K-B$ shares a number of other properties with $R_5$, and we conjecture that it may have the distribution of $R_5$. We also describe the distribution of the convex minorant of a three-dimensional Bessel process with drift.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-23T23:24:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "60G55",
      "60J65"
    ],
    "keywords": [
      "standard one-dimensional brownian motion",
      "three-dimensional bessel process",
      "distribution",
      "surely unique concave majorant",
      "hidden markovian structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}