{
  "id": "0812.0895",
  "version": "v2",
  "published": "2008-12-04T10:02:56.000Z",
  "updated": "2009-01-09T14:10:09.000Z",
  "title": "Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization",
  "authors": [
    "Marek Bozejko",
    "Eugene Lytvynov"
  ],
  "categories": [
    "math.PR",
    "math.OA"
  ],
  "abstract": "Let $T$ be an underlying space with a non-atomic measure $\\sigma$ on it (e.g. $T=\\mathbb R^d$ and $\\sigma$ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of $T$, with freely independent values. Such a process (field), $\\omega=\\omega(t)$, $t\\in T$, is given a rigorous meaning through smearing out with test functions on $T$, with $\\int_T \\sigma(dt)f(t)\\omega(t)$ being a (bounded) linear operator in a full Fock space. We define a set $\\mathbf{CP}$ of all continuous polynomials of $\\omega$, and then define a con-commutative $L^2$-space $L^2(\\tau)$ by taking the closure of $\\mathbf{CP}$ in the norm $\\|P\\|_{L^2(\\tau)}:=\\|P\\Omega\\|$, where $\\Omega$ is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between $L^2(\\tau)$ and a (Fock-space-type) Hilbert space $\\mathbb F=\\mathbb R\\oplus\\bigoplus_{n=1}^\\infty L^2(T^n,\\gamma_n)$, with explicitly given measures $\\gamma_n$. We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set $\\mathbf {CP}$ invariant. (Note that, in the general case, the projection of a continuous monomial of oder $n$ onto the $n$-th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions $\\lambda$ and $\\eta\\ge0$ on $T$, such that, in the $\\mathbb F$ space, $\\omega$ has representation $\\omega(t)=\\di_t^\\dag+\\lambda(t)\\di_t^\\dag\\di_t+\\di_t+\\eta(t)\\di_t^\\dag\\di^2_t$, where $\\di_t^\\dag$ and $\\di_t$ are the usual creation and annihilation operators at point $t$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-01-09T14:10:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-commutative generalized stochastic processes",
      "freely independent values",
      "meixner class",
      "characterization",
      "fock space"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00220-009-0837-x",
      "journal": "Communications in Mathematical Physics",
      "year": 2009,
      "month": "Nov",
      "volume": 292,
      "number": 1,
      "pages": 99
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009CMaPh.292...99B"
    }
  }
}