{
  "id": "1306.6033",
  "version": "v1",
  "published": "2013-06-25T17:01:12.000Z",
  "updated": "2013-06-25T17:01:12.000Z",
  "title": "The Large-$N$ Limits of Brownian Motions on $\\mathbb{GL}_N$",
  "authors": [
    "Todd Kemp"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We introduce a two-parameter family of diffusion processes $(B_{r,s}^N(t))_{t\\ge 0}$, $r,s>0$, on the general linear group $\\mathbb{GL}_N$ that are Brownian motions with respect to certain natural metrics on the group. At the same time, we introduce a two-parameter family of free It\\^o processes $(b_{r,s}(t))_{t\\ge 0}$ in a faithful, tracial $W^\\ast$-probability space, and we prove that the full process $(B^N_{r,s}(t))_{t\\ge 0}$ converges to $(b_{r,s}(t))_{t\\ge 0}$ in noncommutative distribution as $N\\to\\infty$ for each $r,s>0$. The processes $(b_{r,s}(t))_{t\\ge 0}$ interpolate between the free unitary Brownian motion when $(r,s)=(1,0)$, and the free multiplicative Brownian motion when $r=s=\\frac12$; we thus resolve the open problem of convergence of the Brownian motion on $\\mathbb{GL}_N$ posed by Biane in 1997.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-25T17:01:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "46L54",
      "22E30"
    ],
    "keywords": [
      "free unitary brownian motion",
      "free multiplicative brownian motion",
      "general linear group",
      "open problem",
      "natural metrics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.6033K"
    }
  }
}