{
  "id": "2312.01952",
  "version": "v1",
  "published": "2023-12-04T15:11:03.000Z",
  "updated": "2023-12-04T15:11:03.000Z",
  "title": "Fragmentation processes and the convex hull of the Brownian motion in a disk",
  "authors": [
    "Bénédicte Haas",
    "Bastien Mallein"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Motivated by the study of the convex hull of the trajectory of a Brownian motion in the unit disk reflected orthogonally at its boundary, we study inhomogeneous fragmentation processes in which particles of mass $m \\in (0,1)$ split at a rate proportional to $|\\log m|^{-1}$. These processes do not belong to the well-studied family of self-similar fragmentation processes. Our main results characterize the Laplace transform of the typical fragment of such a process, at any time, and its large time behavior. We connect this asymptotic behavior to the prediction obtained by physicists in \\cite{DBBM22} for the growth of the perimeter of the convex hull of a Brownian motion in the disc reflected at its boundary. We also describe the large time asymptotic behavior of the whole fragmentation process. In order to implement our results, we make a detailed study of a time-changed subordinator, which may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-04T15:11:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J25",
      "60D05",
      "60G51",
      "60J80"
    ],
    "keywords": [
      "convex hull",
      "brownian motion",
      "large time asymptotic behavior",
      "self-similar fragmentation processes",
      "large time behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}