Soit %* la fonction conjugue e de % et G %* (N ) l'espace des fonctions holomorphes sur N et qui ve rifient: il existe p # N, m>0 et M 0 telle que:On de montre que la transforme e de Laplace re alise un isomorphisme topologique entre le dual fort de F % (N$) et G %* (N ). On donne ensuite des applications de ce re sultat aÁ l'analyse du bruit blanc. Academic PressLet N$ be the dual of a nuclear Fre chet space N and % a Young function on [0, + [. We define the space F % (N$) of holomorphic functions on N$ which satisfy: for all m>0 and p # N there exists K 0 such thatLet %* be the conjugate function of % and G %* (N ) the space of holomorphic functions on N which satisfy: there exists p # N, m>0 and M 0 such thatWe prove that the Laplace transform realizes a topological isomorphism of the strong dual of F %

A representation of the perturbation series of a general functional measure is given in terms of generalized Feynman graphs and rules. The graphical calculus is applied to certain functional measures of Lévy type. A graphical notion of Wick ordering is introduced and is compared with orthogonal decompositions of the Wiener-Itô-Segal type. It is also shown that the linked cluster theorem for Feynman graphs extends to generalized Feynman graphs. We perturbatively prove existence of the thermodynamic limit for the free energy density and the moment functions. The results are applied to the gas of charged microscopic or mesoscopic particles-neutral in average-in d = 2 dimensions generating a static field with quadratic energy density giving rise to a pair interaction. The pressure function for this system is calculated up to fourth order. We also discuss the subtraction of logarithmically divergent self-energy terms for a gas of only one particle type by a local counterterm of first order.

Duality is established for new spaces of entire functions in two infinite dimensional variables with certain growth rates determined by Young functions. These entire functions characterize the symbols of generalized Fock space operators. As an application, a proper space is found for a solution to a normal-ordered white noise differential equation having highly singular coefficients.Keywords: Fock space; operator symbol; infinite-dimensional holomorphy; normalordered white noise differential equation; quantum white noise.

In this paper we introduce the quadratic Weyl operators canonically associated to the one mode renormalized square of white noise (RSWN) algebra as unitary operator acting on the one mode interacting Fock space {Γ, {ωn, n ∈ ℕ}, Φ} where {ωn, n ∈ ℕ} is the principal Jacobi sequence of the nonstandard (i.e. neither Gaussian nor Poisson) Meixner classes. We deduce the quadratic Weyl relations and construct the quadratic analogue of the Heisenberg Lie group with one degree of freedom. In particular, we determine the manifold structure of the group and introduce a local chart containing the identity on which the group law has a simple rational expression in the chart coordinates (see Theorem 6.3).