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 $\int_T σ(dt)f(t)ω(t)$ being a (bounded) linear operator in a full Fock space. We define a set $\mathbf{CP}$ of all continuous polynomials of $ω$, and then define a con-commutative $L^2$-space $L^2(τ)$ by taking the closure of $\mathbf{CP}$ in the norm $\|P\|_{L^2(τ)}:=\|PΩ\|$, 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 $\mathbb F=\mathbb R\oplus\bigoplus_{n=1}^\infty L^2(T^n,γ_n)$, with explicitly given measures $γ_n$. 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 $λ$ and $η\ge0$ on $T$, such that, in the $\mathbb F$ space, $ω$ has representation $ω(t)=\di_t^†+λ(t)\di_t^†\di_t+\di_t+η(t)\di_t^†\di^2_t$, where $\di_t^†$ and $\di_t$ are the usual creation and annihilation operators at point $t$.