PUB - Publikationen an der Universität Bielefeld

The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space F-ext(H) over H = L-2 (R-d, d sigma) and show how to include the usual Fock space F(H) in it as a subspace. We introduce in F-ext(H) operators a(xi) = integral(Rd) dx xi(x)a(x), xi is an element of S, with a(x) = partial derivative(x)dagger + 2 partial derivative(x)dagger partial derivative(x) + 1 + partial derivative(x) + partial derivative(x)dagger partial derivative(x)partial derivative(x), where partial derivative(x)(dagger) and partial derivative(x) are the creation and annihilation operators at x. We show that (a(xi))(xi is an element of S) is a family of commuting self-adjoint operators in F-ext(H) and construct the Fourier transform in generalized joint eigenvectors of this family. This transform is a unitary I between F-ext(H) and the L-2- space L-2(S', d mu(G)), where mu(G) is the measure of Gamma white noise with intensity sigma. The image of a(xi) under I is the operator of multiplication by [(.),xi], so that a(xi)'s are Gamma field operators. The Fock structure of the Gamma space determined by I coincides with that discovered in Ref. 22. We note that I extends in a natural way the multiple stochastic integral (chaos) decomposition of the "chaotic" subspace of the Gamma space. Next, we introduce and study spaces of test and generalized functions of Gamma white noise and derive explicit formulas for the action of the creation, neutral, and Gamma annihilation operators on these spaces.