{
  "id": "1403.3227",
  "version": "v1",
  "published": "2014-03-13T10:38:09.000Z",
  "updated": "2014-03-13T10:38:09.000Z",
  "title": "A stationary process associated with the Dirichlet distribution arising from the complex projective space",
  "authors": [
    "Nizar Demni"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(U_t)_{t \\geq 0}$ be a Brownian motion valued in the complex projective space $\\mathbb{C}P^{N-1}$. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $|U_t^{1}|^2$ and of $(|U_t^{1}|^2, |U_t^2|^2)$, and express them through Jacobi polynomials in the simplices of $\\mathbb{R}$ and $\\mathbb{R}^2$ respectively. More generally, the distribution of $(|U_t^{1}|^2, \\dots, |U_t^k|^2), 2 \\leq k \\leq N-1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $\\mathcal{U}(N-k+1)$ yet computations become tedious. We also revisit the approach initiated in \\cite{Nec-Pel} and based on a partial differential equation (hereafter pde) satisfied by the Laplace transform of the density. When $k=1$, we invert the Laplace transform and retrieve the expression derived using spherical harmonics. For general $1 \\leq k \\leq N-2$, the integrations by parts performed on the pde lead to a heat equation in the simplex of $\\mathbb{R}^k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-13T10:38:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex projective space",
      "dirichlet distribution arising",
      "stationary process",
      "unitary spherical harmonics",
      "laplace transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.3227D"
    }
  }
}