Meixner Class of Non-Commutative Generalized Stochastic Processes with Freely Independent Values I. A Characterization

Abstract

Let T be an underlying space with a non-atomic measure σ on it (e.g. ${T=\mathbb R^d}$ and σ 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), ω = ω(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 CP of all continuous polynomials of ω, and then define a non-commutative L ²-space L ²(τ) by taking the closure of CP in the norm ${\|P\|_{L^2(\tau)}:=\|P\Omega\|}$ , where Ω is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between L ²(τ) 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 γ _n . We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set CP invariant. (Note that, in the general case, the projection of a continuous monomial of order n onto the n ^th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions λ and η ≥ 0 on T, such that, in the ${\mathbb F}$ space, ω has representation ${\omega(t)=\partial_t^\dagger+\lambda(t)\partial_t^\dagger\partial_t+\partial_t+\eta(t)\partial_t^\dagger\partial^2_t}$ , where ${\partial_t^\dagger}$ and ∂ _t are the usual creation and annihilation operators at point t.