We investigate the finite-time stability of stochastic linear fractional systems with delay (SLFS... more We investigate the finite-time stability of stochastic linear fractional systems with delay (SLFSD) for α∈(1/2,1). By the Banach fixed-point theorem and classical stochastic techniques, the finite-time stability (FTS) of the solution for SLFSD is studied. One example is given to illustrate our theory.
The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most usefu... more The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most useful operators for modeling nonlocal behaviors by fractional differential equations. In terms of Mittag–Leffler function and convolution product, using the Laplace transform, we give the exact values of the solutions of the Liouville–Caputo and Riemann–Liouville time fractional evolution equations associated with the Quantum fractional number operator. Therefore, we study the attractiveness of these solutions.
The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most usefu... more The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most useful operators for modeling nonlocal behaviors by fractional differential equations. In terms of Mittag–Leffler function and convolution prod- uct, using the Laplace transform, we give the exact values of the solutions of the Liouville–Caputo and Riemann–Liouville time fractional evolution equations associated with the Quantum fractional number operator. Therefore, we study the attractiveness of these solutions.
Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class o... more Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class of Hadamard Fractional Itô–Doob Stochastic integral equations (HFIDSIE) of order via the fixed point technique (FPT). Hyers–Ulam stability (HUS) is investigated for HFIDSIE according to the Gronwall inequality. Two theoretical examples are provided to illustrate our results.
Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class o... more Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class of Hadamard Fractional Itô–Doob Stochastic integral equations (HFIDSIE) of order
via the fixed point technique (FPT). Hyers–Ulam stability (HUS) is investigated for HFIDSIE according to the Gronwall inequality. Two theoretical examples are provided to illustrate our results.
By means of the Laplace transform, we give the solution of the generalized Riemann-Liouville and ... more By means of the Laplace transform, we give the solution of the generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations in infinite dimensions associated to the number operator. These solutions are given in terms of the Mittag-Leffler function and the convolution product.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2021
Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is... more Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is the topological dual space of the space of entire functions with exponential growth of order [Formula: see text] and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in [Formula: see text]. These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2020
In this paper, we provide a new reformulation of the quadratic analogue of the Weyl relations. Es... more In this paper, we provide a new reformulation of the quadratic analogue of the Weyl relations. Especially, we offer some adjustments [10] on these relations and the corresponding group law, i.e., the quadratic Heisenberg group law. We provide a much more transparent description of the underlying manifold and we give a connection with the projective group PSU(1,1). Finally, we deduce such a holomorphic realization of the quadratic Heisenberg group.
Bulletin of the Malaysian Mathematical Sciences Society, 2020
In this paper, we introduce a space of θ-admissible distributions denoted by A * θ as well as the... more In this paper, we introduce a space of θ-admissible distributions denoted by A * θ as well as the notion of θ-admissible operators. We study the regularity properties of the classical conditional expectation acting on A * θ and acting on L(A θ , A * θ) which is the space of linear continuous operators from A θ into A * θ. An integral representation with respect to the coordinate system of the quantum white noise (QWN) derivatives and their adjoints {D ± t , D ± * t , t ∈ R} of such conditional expectation is given. Then, we give a quantum white noise counterpart of the Clark formula. Finally, we introduce the QWN Hitsuda-Skorokhod integrals. Such integrals are shown to be QWN martingales using a new notion of QWN conditional expectation. Keywords Quantum white noise stochastic process • QWN derivatives • QWN Hitsuda-Skorokhod integral • QWN martingale Mathematics Subject Classification 81S25 • 60H40 • 46A32 • 46G20 • 46F25 Communicated by Keong Lee.
In this paper, we study the fractional number operator as an analog of the finite-dimensional fra... more In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.
Using the Wick derivation operator and the Wick product of elements in a distribution space F * (... more Using the Wick derivation operator and the Wick product of elements in a distribution space F * (S � ℂ) , we introduce the generalized Riccati Wick differential equation as a distribution analogue of the classical Riccati differential equation. The solution of this new equation is given. Finally, we finish this paper by building some applications. Keywords Wick product • Wick derivation • Generalized Riccati Wick differential equation • Space of entire functions with-exponential growth condition of minimal type Communicated by Carlos Tomei.
Communications on Stochastic Analysis, Dec 1, 2014
By a Wick differential equation, we characterize the operator W l,m (f) studied in [1, 3, 9] wher... more By a Wick differential equation, we characterize the operator W l,m (f) studied in [1, 3, 9] where l, m ∈ N ∪ {0} and f ∈ S(R). As an application we give in our setting a new renormalization in order to get the higher powers of white noise. Then, we investigate the commutation relations obtained from the quantum white noise (QWN) derivatives in order to introduce two operators acting on white noise operators, from which we get the higher powers of quantum white noise derivatives and a *-Lie algebra generalizing the renormalized higher power white noise Lie algebra.
St. Petersburg Polytechnical University Journal: Physics and Mathematics, Dec 1, 2017
Based on nuclear algebra of operators acting on spaces of entire functions with θ-exponential gro... more Based on nuclear algebra of operators acting on spaces of entire functions with θ-exponential growth of minimal type, we introduce the quantum generalized Fourier-Gauss transform, the quantum second quantization as well as the quantum generalized Euler operator of which the quantum differential second quantization and the quantum generalized Gross Laplacian are particular examples. Important relation between the quantum generalized Fourier-Gauss transform, the quantum second quantization and the quantum convolution operator is given. Then, using this relation and under some conditions, we investigate the solution of a initial-value problem associated to the quantum generalized Euler operator. More precisely, we show that the aforementioned solution is the composition of a quantum second quantization and a quantum convolution operator.
Communications on Stochastic Analysis, Dec 1, 2012
In this paper we give the integral representation of the power of the quantum white noise (QWN) E... more In this paper we give the integral representation of the power of the quantum white noise (QWN) Euler operator (∆ Q E) ρ , for ρ ∈ N, in terms of the QWN-derivatives {D − t , D + t ; t ∈ R} as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to (∆ Q E) ρ is worked out in the basis of the QWN coordinate system.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2018
By means of infinite-dimensional nuclear spaces, we generalize important results on the represent... more By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.
In this paper we introduce the quantum white noise (QWN) conservation operator N Q acting on nucl... more In this paper we introduce the quantum white noise (QWN) conservation operator N Q acting on nuclear algebra of white noise operators L(F θ (S ′ C (R)), F * θ (S ′ C (R))) endowed with the Wick product. Similarly to the classical case, we give a useful integral representation in terms of the QWNderivatives {D − t , D + t ; t ∈ R} for the QWN-conservation operator from which it follows that the QWN-conservation operator is a Wick derivation. Via this property, a relation with the Cauchy problem associated to the QWN-conservation operator and the Wick differential equation is worked out.
This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based... more This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based on nuclear algebra of white noise operators acting on spaces of entire functions with θ-exponential growth of minimal type. First, we use extended techniques of rotation invariance operators, the commutation relations with respect to the QWN-derivatives and the QWN-conservation operator. Second, we employ the new concept of QWN-convolution operators. As application, we study and characterize the powers of the QWN-Gross Laplacian. As for their associated Cauchy problem it is solved using a QWN-convolution and Wick calculus.
We investigate the finite-time stability of stochastic linear fractional systems with delay (SLFS... more We investigate the finite-time stability of stochastic linear fractional systems with delay (SLFSD) for α∈(1/2,1). By the Banach fixed-point theorem and classical stochastic techniques, the finite-time stability (FTS) of the solution for SLFSD is studied. One example is given to illustrate our theory.
The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most usefu... more The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most useful operators for modeling nonlocal behaviors by fractional differential equations. In terms of Mittag–Leffler function and convolution product, using the Laplace transform, we give the exact values of the solutions of the Liouville–Caputo and Riemann–Liouville time fractional evolution equations associated with the Quantum fractional number operator. Therefore, we study the attractiveness of these solutions.
The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most usefu... more The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most useful operators for modeling nonlocal behaviors by fractional differential equations. In terms of Mittag–Leffler function and convolution prod- uct, using the Laplace transform, we give the exact values of the solutions of the Liouville–Caputo and Riemann–Liouville time fractional evolution equations associated with the Quantum fractional number operator. Therefore, we study the attractiveness of these solutions.
Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class o... more Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class of Hadamard Fractional Itô–Doob Stochastic integral equations (HFIDSIE) of order via the fixed point technique (FPT). Hyers–Ulam stability (HUS) is investigated for HFIDSIE according to the Gronwall inequality. Two theoretical examples are provided to illustrate our results.
Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class o... more Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class of Hadamard Fractional Itô–Doob Stochastic integral equations (HFIDSIE) of order
via the fixed point technique (FPT). Hyers–Ulam stability (HUS) is investigated for HFIDSIE according to the Gronwall inequality. Two theoretical examples are provided to illustrate our results.
By means of the Laplace transform, we give the solution of the generalized Riemann-Liouville and ... more By means of the Laplace transform, we give the solution of the generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations in infinite dimensions associated to the number operator. These solutions are given in terms of the Mittag-Leffler function and the convolution product.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2021
Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is... more Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is the topological dual space of the space of entire functions with exponential growth of order [Formula: see text] and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in [Formula: see text]. These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2020
In this paper, we provide a new reformulation of the quadratic analogue of the Weyl relations. Es... more In this paper, we provide a new reformulation of the quadratic analogue of the Weyl relations. Especially, we offer some adjustments [10] on these relations and the corresponding group law, i.e., the quadratic Heisenberg group law. We provide a much more transparent description of the underlying manifold and we give a connection with the projective group PSU(1,1). Finally, we deduce such a holomorphic realization of the quadratic Heisenberg group.
Bulletin of the Malaysian Mathematical Sciences Society, 2020
In this paper, we introduce a space of θ-admissible distributions denoted by A * θ as well as the... more In this paper, we introduce a space of θ-admissible distributions denoted by A * θ as well as the notion of θ-admissible operators. We study the regularity properties of the classical conditional expectation acting on A * θ and acting on L(A θ , A * θ) which is the space of linear continuous operators from A θ into A * θ. An integral representation with respect to the coordinate system of the quantum white noise (QWN) derivatives and their adjoints {D ± t , D ± * t , t ∈ R} of such conditional expectation is given. Then, we give a quantum white noise counterpart of the Clark formula. Finally, we introduce the QWN Hitsuda-Skorokhod integrals. Such integrals are shown to be QWN martingales using a new notion of QWN conditional expectation. Keywords Quantum white noise stochastic process • QWN derivatives • QWN Hitsuda-Skorokhod integral • QWN martingale Mathematics Subject Classification 81S25 • 60H40 • 46A32 • 46G20 • 46F25 Communicated by Keong Lee.
In this paper, we study the fractional number operator as an analog of the finite-dimensional fra... more In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.
Using the Wick derivation operator and the Wick product of elements in a distribution space F * (... more Using the Wick derivation operator and the Wick product of elements in a distribution space F * (S � ℂ) , we introduce the generalized Riccati Wick differential equation as a distribution analogue of the classical Riccati differential equation. The solution of this new equation is given. Finally, we finish this paper by building some applications. Keywords Wick product • Wick derivation • Generalized Riccati Wick differential equation • Space of entire functions with-exponential growth condition of minimal type Communicated by Carlos Tomei.
Communications on Stochastic Analysis, Dec 1, 2014
By a Wick differential equation, we characterize the operator W l,m (f) studied in [1, 3, 9] wher... more By a Wick differential equation, we characterize the operator W l,m (f) studied in [1, 3, 9] where l, m ∈ N ∪ {0} and f ∈ S(R). As an application we give in our setting a new renormalization in order to get the higher powers of white noise. Then, we investigate the commutation relations obtained from the quantum white noise (QWN) derivatives in order to introduce two operators acting on white noise operators, from which we get the higher powers of quantum white noise derivatives and a *-Lie algebra generalizing the renormalized higher power white noise Lie algebra.
St. Petersburg Polytechnical University Journal: Physics and Mathematics, Dec 1, 2017
Based on nuclear algebra of operators acting on spaces of entire functions with θ-exponential gro... more Based on nuclear algebra of operators acting on spaces of entire functions with θ-exponential growth of minimal type, we introduce the quantum generalized Fourier-Gauss transform, the quantum second quantization as well as the quantum generalized Euler operator of which the quantum differential second quantization and the quantum generalized Gross Laplacian are particular examples. Important relation between the quantum generalized Fourier-Gauss transform, the quantum second quantization and the quantum convolution operator is given. Then, using this relation and under some conditions, we investigate the solution of a initial-value problem associated to the quantum generalized Euler operator. More precisely, we show that the aforementioned solution is the composition of a quantum second quantization and a quantum convolution operator.
Communications on Stochastic Analysis, Dec 1, 2012
In this paper we give the integral representation of the power of the quantum white noise (QWN) E... more In this paper we give the integral representation of the power of the quantum white noise (QWN) Euler operator (∆ Q E) ρ , for ρ ∈ N, in terms of the QWN-derivatives {D − t , D + t ; t ∈ R} as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to (∆ Q E) ρ is worked out in the basis of the QWN coordinate system.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2018
By means of infinite-dimensional nuclear spaces, we generalize important results on the represent... more By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.
In this paper we introduce the quantum white noise (QWN) conservation operator N Q acting on nucl... more In this paper we introduce the quantum white noise (QWN) conservation operator N Q acting on nuclear algebra of white noise operators L(F θ (S ′ C (R)), F * θ (S ′ C (R))) endowed with the Wick product. Similarly to the classical case, we give a useful integral representation in terms of the QWNderivatives {D − t , D + t ; t ∈ R} for the QWN-conservation operator from which it follows that the QWN-conservation operator is a Wick derivation. Via this property, a relation with the Cauchy problem associated to the QWN-conservation operator and the Wick differential equation is worked out.
This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based... more This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based on nuclear algebra of white noise operators acting on spaces of entire functions with θ-exponential growth of minimal type. First, we use extended techniques of rotation invariance operators, the commutation relations with respect to the QWN-derivatives and the QWN-conservation operator. Second, we employ the new concept of QWN-convolution operators. As application, we study and characterize the powers of the QWN-Gross Laplacian. As for their associated Cauchy problem it is solved using a QWN-convolution and Wick calculus.
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Papers by Hafedh Rguigui
via the fixed point technique (FPT). Hyers–Ulam stability (HUS) is investigated for HFIDSIE according to the Gronwall inequality. Two theoretical examples are provided to illustrate our results.
via the fixed point technique (FPT). Hyers–Ulam stability (HUS) is investigated for HFIDSIE according to the Gronwall inequality. Two theoretical examples are provided to illustrate our results.