WHITE NOISE LÉVY–MEIXNER PROCESSES THROUGH A TRANSFER PRINCIPAL FROM ONE-MODE TO ONE-MODE TYPE INTERACTING FOCK SPACES

Abstract

Consider the Lévy–Meixner one-mode interacting Fock space {ΓLM, 〈 ⋅, ⋅ 〉LM}. Inspired by a derivative formula appearing in 〈 ⋅, ⋅ 〉LM, we define scalar products 〈 ⋅, ⋅ 〉LM , nin symmetric n-particle spaces. Then, we introduce a class of one-mode type interacting Fock spaces [Formula: see text] naturally associated to the one-dimensional infinitely divisible distributions with Lévy–Meixner type {μr; r > 0}. The Fourier transform in generalized joint eigenvectors of a family [Formula: see text] of Lévy–Meixner Jacobi fields provides a way to explicit a unitary isomorphism 𝔘LMbetween [Formula: see text] and the so-called Lévy–Meixner white noise space [Formula: see text]. We derive a chaotic decomposition property of the quadratic integrable functionals of the Lévy–Meixner white noise processes in terms of an appropriate Wick tensor product. For their stochastic regularity, we give explicit form and sharp estimate of the associated Donsker's delta function.