Interacting Fock Expansion of Lévy White Noise Functionals*

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Abstract

Consider the Lévy white noise space (&*,ℬ(&*),μ), where &* is the Schwartz distributions over ℝd and μ is a Lévy white noise measure lifted from a 1-dimensional infinitely divisible distribution with finite moments. We give explicit forms and recursion formulas of moment and renormalization kernels for the Lévy white noise measure. By defining inner products (.,.)[n] in n-particle spaces, we establish an interacting Fock space ⊕n=0∞ℋ(n) and the interacting Fock expansions for Lévy white noise functionals. The usual Fock space Γ(H)=⊕n=0∞H⊗n can be viewed as a quotient space of the interacting Fock space. As a particular case, we give the interacting Fock expansion for gamma white noise functionals.

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