{
  "id": "1302.4844",
  "version": "v2",
  "published": "2013-02-20T09:19:38.000Z",
  "updated": "2013-04-07T09:04:27.000Z",
  "title": "General position of a projection and its image under a free unitary Brownian motion",
  "authors": [
    "Nizar Demni",
    "Taoufik Hmidi"
  ],
  "comment": "The letter `a' was used to denote two different objects: an operator and a real number. The operator is now denoted `S' referring to `symmetry'",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "Given an orthogonal projection $P$ and a free unitary Brownian motion $Y = (Y_t)_{t \\geq 0}$ in a $W^{\\star}$-non commutative probability space such that $Y$ and $P$ are $\\star$-free in Voiculescu's sense, the main result of this paper states that $P$ and $Y_tPY_t^{\\star}$ are in general position at any time $t$. To this end, we study the dynamics of the unitary operator $SY_tSY_t^{\\star}$ where $S = 2P-1$. More precisely, we derive a partial differential equation for the Herglotz transform of its spectral distribution, say $\\mu_t$. Then, we provide a flow on the interval $[-1,1]$ in such a way that the Herglotz transform of $\\mu_t$ composed with this flow is governed by both the Herglotz transforms of the initial ($t=0$) and the stationary ($t = \\infty)$ distributions. This fact allows to compute the weight that $\\mu_t$ assigns to $z=1$ leading to the main result. As a by-product, the weight that the spectral distribution of the free Jacobi process assigns to $x=1$ follows after a normalization. In the last part of the paper, we use combinatorics of non crossing partitions in order to analyze the term corresponding to the exponential decay $e^{-nt}$ in the expansion of the $n$-th moment of $SY_tSY_t^{\\star}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-04-07T09:04:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "35F20"
    ],
    "keywords": [
      "free unitary brownian motion",
      "general position",
      "herglotz transform",
      "projection",
      "spectral distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.4844D"
    }
  }
}