Abdessatar Barhoumi

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.

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.

White noise quantum time shifts

Abstract In the present paper, we extend the notion of quantum time shift, and the related results obtained in [Accardi, Barhoumi and Ouerdiane, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9: 215–248, 2006], from representations of current algebras of the Heisenberg Lie algebra to representations of current algebras of the Oscillator Lie algebra. This produces quantum extensions of a class of classical Lévy processes much wider than the usual Brownian motion. In particular, this class of processes includes the Meixner processes and, by an approximation procedure, we construct quantum extensions of all classical Lévy processes with a Lévy measure with finite variance. Finally, we compute the explicit form of the action, on the Weyl operators of the initial space, of the generators of the quantum Markov processes canonically associated to the above class of Lévy processes. The emergence of the Meixner classes in connection with the renormalized second order white noise, is now well known. The fact that they also emerge from first order noise in a simple and canonical way comes somehow as a surprise.

Jacobi sequences of squares of random variables

Abstract We express the Jacobi sequences of the square of a real valued random variable with all moments, not necessarily symmetric, as functions of the corresponding sequences of the random variable itself. In the symmetric case, the result is known and, we give a short, purely algebraic proof of it. We apply our result to the square of the Gamma distribution, i.e. the 4th power of the standard Gaussian. The result confirms the conjecture that belongs to the polynomial class, but its principal Jacobi sequence grows like , not as expected.

Nuclear Realization of Virasoro–Zamolodchikov-w∞ ⋆-Lie Algebras Through the Renormalized Higher Powers of Quantum Meixner White Noise

By using an appropriate one-mode type interacting Fock spaces, , introduced in [1], we define a nuclear triple of test and generalized functions, with θ being a suitable Young function. Moreover, we prove general characterization theorems for the fundamental nuclear spaces. For the applications, we introduce new renormalized products for the generators of the renormalized higher powers of white noise ⋆-Lie algebra and the Virasoro-Zamolodchikov-w∞ ⋆-Lie algebra. Then we show that these new renormalized products lead to nuclear realizations of these Lie algebras in terms of quantum Meixner white noise operators.

*–Lie Algebras Canonically Associated to Probability Measures on {\pmb {\varvec{\mathbb {R}}}} with All Moments

In the paper Accardi et al.: Identification of the theory of orthogonal polynomials in d–indeterminates with the theory of 3–diagonal symmetric interacting Fock spaces on \(\mathbb {C} ^d\), submitted to: IDA–QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), [1], it has been shown that, with the natural definitions of morphisms and isomorphisms (that will not be recalled here) the category of orthogonal polynomials in a finite number of variables is isomorphic to the category of symmetric interacting fock spaces (IFS) with a 3–diagonal structure. Any IFS is canonically associated to a \(*\)–Lie algebra (commutation relations) and a \(*\)–Jordan algebra (anti–commutation relations). In this paper we continue the study of these algebras, initiated in Accardi et al. An Information Complexity index for Probability Measures on \(\mathbb {R}\) with all moments, submitted to: IDA–QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), [2], in the case of polynomials in one variable, refine the definition of information complexity index of a probability measure on the real line, introduced there, and prove that the \(*\)–Lie algebra canonically associated to the probability measures of complexity index (0, K, 1), defining finite–dimensional approximations, in the sense of Jacobi sequences, of the Heisenberg algebra, coincides with the algebra of all \(K \times K\) complex matrices.