CONTENTS -- Preface -- A Quarter Century of White Noise Theory -- 1. Pre-theory period (white noise without a theory) -- 2. Creation of white noise theory (1975) -- 3. Kubo-Takenaka's construction (1980) -- 4. Characterization theorems of Potthoff-Streit and Lee (1991) -- 5. More spaces of generalized functions (1992-2000) -- 6. Applications of white noise theory -- 7. References on white noise theory -- References -- Integral Transform and Segal-Bargmann Representation Associated to q-Charlier Polynomials -- 1 Introduction -- 2 q-deformed Charlier Polynomials -- 3 Segal-Bargmann Transform and Hardy Space -- 4 Main Results -- Appendix -- References -- Notions of Independence in Algebraic Probability Theory and Set Partition Statistics -- 1 Singleton condition and set partition statistics -- 2 Law of large numbers central limit theorems and entangled ergodicity -- 3 Set partition statistics on discrete groups -- 4 Singleton independence -- 5 A limit theorem on the Haagerup states -- 6 Quantum decomposition in a discrete group -- 7 Interacting Fock spaces associated with a discrete group -- 8 Applications to limit theorems in algebraic probability theory -- References -- Some Thoughts on the Infinite Dimensional Harmonic Analysis -- 1 Background -- 2 Roles of whiskers -- 3 Conformal group -- 4 Concluding remarks -- References -- A Treatment of Quantum Baker's Map by Chaos Degree -- 1 Introduction -- 2 Quantum vs. Classical Dynamics -- 3 Coherent States for the Quantum Baker's Map -- 4 Chaos Degree -- 5 Chaos Degree for the Quantum Baker's map -- 6 Numerical Simulation of the Chaos Degree and Classical-Quantum Correspondence -- References -- Poisson Noise Analysis Based on the Levy Laplacian -- 1 Introduction -- 2 Preliminaries -- 3 The Levy Laplacian acting on Poisson noise functionals.

4 A semi-group generated by the Levy Laplacian -- 5 Stochastic expressions of the semigroup generated by the Levy Laplacian -- References -- A Hausdorff-Young Inequality for White Noise Analysis -- 1 INTRODUCTION -- 2 BACKGROUND -- 3 HAUSDORFF-YOUNG INEQUALITY FOR WHITE NOISE ANALYSIS -- REFERENCES -- The Clark Formula of Generalized Wiener Functionals -- 1 Introduction -- 2 A Gel'fand triple associated with the classical Wiener space -- 3 The spaces of test and generalized functions -- 4 The Clark formula for generalized Wiener functionals -- References -- Inverse S-Transform Wick Product and Overcompleteness of Exponential Vectors -- Introduction -- 1 White Noise Functions -- 2 Complex White Noise -- 3 Representation of White Noise Functions -- 4 Representation of White Noise Operators -- 5 Wick Product and Overcompleteness of Exponential Vectors -- Appendix: Multi-Variable White Noise Functions -- References -- Topics on Complex Gaussian Random Fields -- 0 Introduction -- 1 Background -- 2 Complex Gaussian random field -- 3 Innovation -- References -- Quantum Information in Space and Time and Theory of Stochastic Processes -- 1 Introduction -- 2 Bell's Theorem -- 3 Localized Detectors -- 4 The Quantum Key Distribution -- 5 Conclusions -- References.