### À propos des distributions sur l'espace de Wiener

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We prove that derivatives of any finite order of Donsker's delta functionals are well-defined elements in the space of Hida distributions. We also show the convergence to the derivative of Donsker's delta functionals of two different approximations. Finally, we present an existence result of finite product and infinite series of the derivative of the Donsker delta functionals.

2000 Mathematics Subject Classification: Primary 46F25, 26A33; Secondary: 46G20In this paper we study the generalized Riemann-Liouville (resp. Caputo) time fractional evolution equation in infinite dimensions. We show that the explicit solution is given as the convolution between the initial condition and a generalized function related to the Mittag-Leffler function. The fundamental solution corresponding to the Riemann-Liouville time fractional evolution equation does not admit a probabilistic...

he paper is devoted to investigation of Gegenbauer white noise functionals. A particular attention is paid to the construction of the infinite dimensional Gegenbauer white noise measure ${}_{\beta}$, via the Bochner-Minlos theorem, on a suitable nuclear triple. Then we give the chaos decomposition of the L²-space with respect to the measure ${}_{\beta}$ by using the so-called β-type Wick product.

We consider the multiplicative algebra P(𝒢₊') of continuous scalar polynomials on the space 𝒢₊' of Roumieu ultradistributions on [0,∞) as well as its strong dual P'(𝒢₊'). The algebra P(𝒢₊') is densely embedded into P'(𝒢₊') and the operation of multiplication possesses a unique extension to P'(𝒢₊'), that is, P'(𝒢₊') is also an algebra. The operation of differentiation on these algebras is investigated. The polynomially extended Laplace transformation and its connections with the differentiation...

In this paper, we study a representation of the quantum Itô algebra in Fock space and then by using a noncommutative Radon-Nikodym type theorem we study the density operators of output states as quantum martingales, where the output states are absolutely continuous with respect to an input (vacuum) state. Then by applying quantum martingale representation we prove that the density operators of regular, absolutely continuous output states belong to the commutant of the ⋆-algebra parameterizing the...

We study a quantum extension of the Lévy Laplacian, so-called quantum Lévy-type Laplacian, to the nuclear algebra of operators on spaces of entire functions. We give several examples of the action of the quantum Lévy-type Laplacian on basic operators and we study a quantum white noise convolution differential equation involving the quantum Lévy-type Laplacian.

The purpose of this paper is to give an illustration of results on integrability of distributions and orbits of vector fields on Banach manifolds obtained in [5] and [4]. Using arguments and results of these papers, in the context of a separable Hilbert space, we give a generalization of a Theorem of accessibility contained in [3] and [6] for articulated arms and snakes in a finite dimensional Hilbert space.

The paper gives a new application of the white noise distribution theory via a proof of irreducibility of the energy representation of a group of ${C}^{\infty}$-maps from a compact Riemann manifold to a semi-simple compact Lie group.