{
  "id": "2104.07859",
  "version": "v1",
  "published": "2021-04-16T02:30:06.000Z",
  "updated": "2021-04-16T02:30:06.000Z",
  "title": "The Brown measure of a family of free multiplicative Brownian motions",
  "authors": [
    "Brian C. Hall",
    "Ching-Wei Ho"
  ],
  "comment": "61 pages, 16 figures",
  "categories": [
    "math.PR",
    "hep-th",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider a family of free multiplicative Brownian motions $b_{s,\\tau}$ parametrized by a positive real number $s$ and a nonzero complex number $\\tau$ satisfying $\\left\\vert \\tau-s\\right\\vert \\leq s,$ with an arbitrary unitary initial condition. We compute the Brown measure $\\mu_{s,\\tau}$ of $b_{s,\\tau}$ and find that it has a simple structure, with a density in logarithmic coordinates that is constant in the $\\tau$-direction. We also find that all the Brown measures with $s$ fixed and $\\tau$ varying are related by pushforward under a natural family of maps. Our results generalize those of Driver-Hall-Kemp and Ho-Zhong for the case $\\tau=s.$ We use a version of the PDE method introduced by Driver-Hall-Kemp, but with some significant technical differences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-16T02:30:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "46L54"
    ],
    "keywords": [
      "free multiplicative brownian motions",
      "brown measure",
      "arbitrary unitary initial condition",
      "nonzero complex number",
      "significant technical differences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}