Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization

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 $\mathbf{C}\mathbf{P}$ of all continuous polynomials of $\omega$, and then define a con-commutative $L^2$-space $L^2(\tau )$ by taking the closure of $\mathbf{C}\mathbf{P}$ in the norm $\mathrm{\parallel }P{\mathrm{\parallel }}_{L^2(\tau )}:=\mathrm{\parallel }P\mathrm{\Omega }\mathrm{\parallel }$, where $\mathrm{\Omega }$ 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 $\mathbb{F}=\mathbb{R}\oplus {⨁}_{n=1}^{\mathrm{\infty }}{L}^2(T^n,{\gamma }_n)$, 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 $\eta \ge 0$ on $T$, such that, in the $\mathbb{F}$ space, $\omega$ has representation $\omega (t)={\text{di}}_t^{\mathrm{†}}+\lambda (t){\text{di}}_t^{\mathrm{†}}{\text{di}}_t+{\text{di}}_t+\eta (t){\text{di}}_t^{\mathrm{†}}{\text{di}}_t^2$, where ${\text{di}}_t^{\mathrm{†}}$ and ${\text{di}}_t$ are the usual creation and annihilation operators at point $t$.