Habib Ouerdiane

ANALYTIC CHARACTERIZATION OF GENERALIZED FOCK SPACE OPERATORS AS TWO-VARIABLE ENTIRE FUNCTIONS WITH GROWTH CONDITION

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.

A QUANTUM APPROACH TO LAPLACE OPERATORS

In this paper, a theory of stochastic processes generated by quantum extensions of Laplacians is developed. Representations of the associated heat semigroups are discussed by means of suitable time shifts. In particular the quantum Brownian motion associated to the Levy–Laplacian is obtained as the usual Volterra–Gross Laplacian using the Cesaro Hilbert space as initial space of our process as well as multiplicity space of the associated white noise.

QWN-EULER OPERATOR AND ASSOCIATED CAUCHY PROBLEM

In this paper the quantum white noise (QWN)-Euler operator is defined as the sum , where and NQ stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that has an integral representation in terms of the QWN-derivatives as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.

Convolution calculus on white noise spaces and Feynman graph representation of generalized renormalization flows

In this note we outline some novel connections between the following fields: 1) Convolution calculus on white noise spaces 2) Pseudo-differential operators and Levy processes on infinite dimensional spaces 3) Feynman graph representations of convolution semigroups 4) generalized renormalization group flows and 5) the thermodynamic limit of particle systems.

Expansion in series of exponential polynomials of mean-periodic functions

Let θ be a Young function and consider the space ℱθ(ℂ) of all entire functions on ℂ with θ-exponential growth. In this article, we are interested in the solutions f ∈ ℱθ(ℂ) of the convolution equation T ⋆ f = 0, called mean-periodic functions, where T is in the topological dual of ℱθ(ℂ). We show that each mean-periodic function admits an expansion as a convergent series of exponential polynomials.

A NOTE ON CONVOLUTION OPERATORS IN WHITE NOISE CALCULUS

We derive some characteristic properties of the convolution operator acting on white noise functions and prove that the convolution product of white noise distributions coincides with their Wick product. Moreover, we show that the S-transform and the Laplace transform coincide on Fock space.

On the Quadratic Heisenberg Group

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).

Pascal white noise calculus

In this paper white noise analysis with respect to the Lévy process with negative binomial distributed marginals is investigated. An appropriate space of distributions, ℰ ′, is used to describe the structure of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure ΛNB. The constructed decomposition is used to define a nuclear triple of test and generalized functions, where θ is a Young function satisfying some suitable conditions. By using the 𝒮-transform and the symbol transform σNB, a general characterization theorems are proven for Pascal white noise distributions, white noise test functions and white noise operators in terms of analytical functions with growth condition of exponential type. As application, some quantum stochastic differential equations are solved with special emphasis on Wick calculus.

UNITARY REPRESENTATIONS OF THE WITT AND sl(2, ℝ)-ALGEBRAS THROUGH RENORMALIZED POWERS OF THE QUANTUM PASCAL WHITE NOISE

By using an appropriate space of distributions, , we derive the chaos decomposition property of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure ΛNB. The constructed decomposition is used to define a nuclear triple of test and generalized functions, where θ is a Young function satisfying some suitable conditions. A general characterization theorems are proven for the Pascal white noise distributions, white noise test functions and white noise operators in terms of analytical functions with growth condition of exponential type. By using appropriate renormalization procedure, we obtain the representation of the square of white noise obtained by Accardi–Franz–Skeide in Ref. 5. Finally, we investigate the main aim of this paper which is to give unitary equivalent representations of the Witt algebra in the basis of Pascal white noise theory.

Generic Quantum Markov Semigroups: the Fock Case

We introduce the class of generic quantum Markov semigroups. Within this class we study the class corresponding to the Fock case which is further split into four subclasses each of which contains both bounded and unbounded generators, depending on some global characteristics of the intensities of jumps. For the first two of these classes we find an explicit solution which reduces the problem of finding the quantum semigroup to the calculation of two classical semigroups, one of which is diagonal (in a suitable basis) and the other one is triangular (in the same basis). In the bounded case our formula gives the unique solution. In the unbounded case it gives one solution, which we conjecture to be the minimal one.