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 s on it (e. g. T = 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 is an element of T, is given a rigorous meaning through smearing out with test functions on T, with integral(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 omega, and then define a non-commutative L-2-space L-2(tau) by taking the closure of CP in the norm parallel to P parallel to(L2(tau)) := parallel to P Omega parallel to, 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 F = R circle plus circle plus(infinity)(n=1) 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 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 lambda and eta >= 0 on T, such that, in the F space, omega has representation omega(t) = partial derivative(dagger)(t) + lambda(t)partial derivative(dagger)(t)partial derivative(t) + partial derivative(t) + eta(t)partial derivative(dagger)(t)partial derivative(2)(t), where partial derivative(dagger)(t) and partial derivative(t) are the usual creation and annihilation operators at point t.