Preface | p. vii |
Introduction | p. 1 |
Preliminaries | p. 1 |
Our idea of establishing white noise analysis | p. 2 |
A brief synopsis of the book | p. 6 |
Some general background | p. 8 |
Characteristics of white noise analysis | p. 10 |
Generalized white noise functionals | p. 13 |
Brownian motion and Poisson process; elemental stochastic processes | p. 13 |
Comparison between Brownian motion and Poisson process | p. 21 |
The Bochner-Minlos theorem | p. 22 |
Observation of white noise through the Levy's construction of Brownian motion | p. 26 |
Spaces (L[superscript 2]), F and F arising from white noise | p. 27 |
Generalized white noise functionals | p. 35 |
Creation and annihilation operators | p. 50 |
Examples | p. 54 |
Addenda | p. 57 |
Elemental random variables and Gaussian processes | p. 63 |
Elemental noises | p. 63 |
Canonical representation of a Gaussian process | p. 70 |
Multiple Markov Gaussian processes | p. 81 |
Fractional Brownian motion | p. 86 |
Stationarity of fractional Brownian motion | p. 91 |
Fractional order differential operator in connection with Levy's Brownian motion | p. 95 |
Gaussian random fields | p. 97 |
Linear processes and linear fields | p. 99 |
Gaussian systems | p. 100 |
Poisson systems | p. 107 |
Linear functionals of Poisson noise | p. 108 |
Linear processes | p. 109 |
Levy field and generalized Levy field | p. 113 |
Gaussian elemental noises | p. 114 |
Harmonic analysis arising from infinite dimensional rotation group | p. 115 |
Introduction | p. 115 |
Infinite dimensional rotation group O(E) | p. 117 |
Harmonic analysis | p. 120 |
Addenda to the diagram | p. 126 |
The Levy group, the Windmill subgroup and the sign-changing subgroup of O(E) | p. 128 |
Classification of rotations in O(E) | p. 136 |
Unitary representation of the infinite dimensional rotation group O(E) | p. 139 |
Laplacian | p. 140 |
Complex white noise and infinite dimensional unitary group | p. 153 |
Why complex? | p. 153 |
Some background | p. 154 |
Subgroups of U(E[subscript c]) | p. 159 |
Applications | p. 170 |
Characterization of Poisson noise | p. 175 |
Preliminaries | p. 175 |
A characteristic of Poisson noise | p. 178 |
A characterization of Poisson noise | p. 186 |
Comparison of two noises; Gaussian and Poisson | p. 191 |
Poisson noise functionals | p. 194 |
Innovation theory | p. 197 |
A short history of innovation theory | p. 198 |
Definitions and examples | p. 200 |
Innovations in the weak sense | p. 204 |
Some other concrete examples | p. 208 |
Variational calculus for random fields and operator fields | p. 211 |
Introduction | p. 211 |
Stochastic variational equations | p. 212 |
Illustrative examples | p. 213 |
Integrals of operators | p. 216 |
Operators of linear form | p. 216 |
Operators of quadratic forms of the creation and the annihilation operators | p. 217 |
Polynomials in [partial differential subscript t], [partial differential]*[subscript s]; t, s [set membership] R, of degree 2 | p. 220 |
Four notable roads to quantum dynamics | p. 223 |
White noise approach to path integrals | p. 223 |
Hamiltonian dynamics and Chern-Simons functional integral | p. 230 |
Dirichlet forms | p. 234 |
Time operator | p. 239 |
Addendum: Euclidean fields | p. 248 |
Appendix | p. 249 |
Bibliography | p. 253 |
Subject Index | p. 263 |
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