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• W3100805935 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), $${tin T}$$ , 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 $${|P|_{L^2(tau)}:=|POmega|}$$ , 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 $${mathbb F=mathbb Roplusbigoplus_{n=1}^infty L^2(T^n,gamma_n)}$$ , 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 $${omega(t)=partial_t^dagger+lambda(t)partial_t^daggerpartial_t+partial_t+eta(t)partial_t^daggerpartial^2_t}$$ , where $${partial_t^dagger}$$ and ∂ t are the usual creation and annihilation operators at point t." @default.
• W3100805935 created "2020-11-23" @default.
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• W3100805935 date "2009-05-22" @default.
• W3100805935 modified "2023-09-26" @default.
• W3100805935 title "Meixner Class of Non-Commutative Generalized Stochastic Processes with Freely Independent Values I. A Characterization" @default.
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• W3100805935 doi "https://doi.org/10.1007/s00220-009-0837-x" @default.
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