CONTENTS -- Preface -- On the Central Extensions of the Heisenberg Algebra L. Accardi & A. Boukas -- 1. Central extensions of Lie algebras -- 2. Central extensions of the Heisenberg algebra -- 3. Representations of CE(Heis) -- 4. Random variables associated with CE(Heis) -- 5. The centrally extended Heisenberg group -- References -- Representations of the Levy-Meixner Oscillator Algebra and the Overcompleteness of the Associated Sequences of Coherent States A. Barhoumi, H. Ouerdiane & A. Riahi -- 1. Introduction -- 2. The Levy-Meixner oscillator algebra -- 3. One-mode interacting Fock space representation of the L¶evy-Meixner oscillator -- 4. Levy-Meixner Coherent states -- 4.1. Bargmann space and holomorphic representatio -- 4.2. The coherent vectors z -- 4.3. Overcompletness of the Levy-Meixner coherent vectors -- 5. Complex Hypercontractivity for the Meixner class -- References -- Some Systems of Dualities in White Noise Analysis T. Hida -- 1. Introduction -- 2. Le passage du fini µa l'infinit -- 2.1. Finite dimensional approximations -- 2.2. Infinite dimensional rotation group -- 3. Quadratic functionals of white noise -- 4. Duality in the space of quadratic generalized functionals -- 5. White noise functionals of higher chaos degree -- References -- Quantum White Noise Derivatives and Associated Differential Equations for White Noise Operators U.C. Ji & N. Obata -- Keywords: -- 1. Introduction -- 2. Quantum white noise derivatives -- 2.1. White noise operators -- 2.2. Quantum white noise derivatives -- 3. Wick derivations -- 4. Differential equations associated with Wick derivations -- 5. White noise operators implementing transformed annihilation and creation operators -- Acknowledgements -- References.

The Gibbs Conditioning Principle for White Noise Distribu- tions: Interacting and Non-Interacting Cases F. Cipriano, S. Gheryani & H. Ouerdiane -- Keywords: -- 1. Introduction -- 2. Background -- 2.1. Definitions and results about the space of holomorphic functions with exponential growth -- 2.2. Basic results about Large deviation theory and the Gibbs conditioning principle -- 3. The Non-Interacting case -- 4. The interacting case -- References -- Markov Triplets on CAR Algebras J. Pitrik -- Keywords: -- 1. Introduction -- 2. The CAR algebra -- 3. The strong subadditivity of the von Neumann entropy and the Markov property -- Acknowledgments -- References -- Quantum Fokker-Planck Models: Limiting Case in the Lindblad Condition F. Fagnola & L. Neumann -- Keywords: -- 1. Quantum Fokker-Planck model -- 2. Previous results -- 3. Irreducibility for = 0 -- Acknowledgement: -- References -- Generalized Euler Heat Equation A. Barhoumi, H. Ouerdiane & H. Rguigui -- Keywords: -- 1. Introduction -- 2. Preliminaries -- 3. The GA, B-transform -- 4. Generalized Euler operator -- 5. Generalized Euler heat equation -- References -- On Quantum De Finetti's Theorems V. Crismale & Y.G. Lu -- 1. Preliminaries -- 2. Singleton and exchangeability conditions -- 3. De Finetti type results -- 4. Final remarks -- References -- Kolmogorovian Model for EPR-Experiment D. Avis, P. Fischer, A. Hilbert & A. Khrennikov -- 1. Introduction -- 2. CHSH inequality -- 3. Proper random experiment -- 4. Frequency analysis -- 5. Consistent joint probability space -- 6. Two-valued random variables -- References -- Free White Noise Stochastic Equation L. Accardi, W. Ayed & H. Ouerdiane -- 1. Introduction -- 2. Fock operator valued distributions -- 3. Free Fock white noise -- 4. Free white noise integrals -- 4.1. Estimates on the free white noise stochastic integrals.

4.2. Free White Noise Stochastic Equations -- 5. Normal Form of the Free White Noise Equations -- References -- Levy Models Robustness and Sensitivity F. E. Benth, G. Di Nunno & A. Khedher -- 1. Introduction -- 2. Density based methods for the computation of derivatives -- 2.1. Some mathematical preliminaries -- 2.2. The density method -- 2.3. The conditional density method -- 3. Robustness of the delta to model choice -- 3.1. Small jump approximations of Levy processes -- 3.2. Robustness to model choice -- 3.3. Robustness to smoothing of a Levy process -- 3.4. Some numerical issues -- 4. Numerical examples -- 5. Conclusion -- References -- Quantum Heat Equation with Quantum K-Gross Laplacian: Solutions and Integral Representation S. Horrigue & H. Ouerdiane -- Keywords: -- 1. Introduction -- 2. Preliminaries -- 2.1. Space of entire functions with growth condition -- 2.2. Operator theory -- 2.3. Integral Kernel Operators and Expansion Theorem -- 3. Quantum K-Gross Laplacian -- 4. Quantum white noise derivatives of quantum stochastic integrals -- 4.1. Gross derivative -- 4.2. Quantum stochastic integral of the quantum Gross Laplacian -- 5. Quantum Cauchy problem associated to the K-Gross Laplacian -- 5.1. Solution of the Quantum Cauchy problem associated to the K-Gross Laplacian -- 5.2. Integral representation of the solution -- 6. Remarks -- References -- On Marginal Markov Processes of Quantum Quadratic Stochastic Processes F. Mukhamedov -- Keywords: -- 1. Introduction -- 2. Preliminaries -- 3. Marginal Markov Processes and Ergodic Principle -- 3.1. Case type A -- 3.2. Case type B -- Acknowledgement -- References -- On the Applicability of Multiplicative Renormalization Method for Certain Power Functions I. Kubo, H.-H. Kuo & S. Namli -- 1. Multiplicative renormalization method -- 2. Power function of order · = 1/2 -- 3. Power functions of other order.

References -- Convolution Equation: Solution and Probabilistic Representation J. L. Da Silva, M. Erraoui & H. Ouerdiane -- Keywords: -- 1. Introduction -- 2. Preliminaries -- Functional spaces -- Laplace transform -- Convolution -- Operator K and ¤K -- 3. Convolution equation: existence and probabilistic representation -- References -- From Classical to Quantum Entropy Production F. Fagnola & R. Rebolledo -- Keywords: -- 1. Introduction -- 2. Detailed balance and entropy production -- 2.1. The commutative detailed balance condition -- 2.2. Entropy production for classical Markov processes -- 2.3. The quantum detailed balance condition -- 3. Forward and backward two point states -- 4. Entropy production for a QMS -- 5. An example of a quantum 3-level system -- 6. Conclusions and outlook -- Acknowledgments -- References -- Extending the Set of Quadratic Exponential Vectors L. Accardi, A. Dhahri & M. Skeide -- 1. Introduction -- 2. The result -- References -- On Operator-Parameter Transforms Based on Nuclear Alge- bra of Entire Functions and Applications A. Barhoumi, H. Ouerdiane, H. Rguigui & A. Riahi -- Keywords: -- 1. Introduction -- 2. Preliminaries -- 3. The operator-parameter GA, B- and FA, B-transforms -- 4. Generalized Gross heat equation -- References -- Dissipative Quantum Annealing D. de Falco, E. Pertoso & D. Tamascelli -- Keywords: -- 1. Introduction -- 2. Quantum annealing -- 3. Adiabatic quantum computation -- 4. Dissipative SchrÄodinger equation -- 4.1. Continuous case -- 4.2. Discrete case -- 5. Conclusions and Outlook: Stabilized quantum annealing -- References -- Author Index.