Interacting Fock Expansion of Lévy White Noise Functionals

Abstract

Consider the Levy white noise space (S*,ℬ(S*),μ), where S* is the Schwartz distributions over Rd and μ is a Levy 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 Levy 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 Levy white noise functionals. The usual Fock space Γ(H)=⊕n=0∞\(H^{\hat \otimes n} \) 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.