Lectures On White Noise Functionals

By: Takeyuki Hida, Si Si

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Hardcover | 10 July 2008

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White noise analysis is an advanced stochastic calculus that has developed extensively since three decades ago. It has two main characteristics. One is the notion of generalized white noise functionals, the introduction of which is oriented by the line of advanced analysis, and they have made much contribution to the fields in science enormously. The other characteristic is that the white noise analysis has an aspect of infinite dimensional harmonic analysis arising from the infinite dimensional rotation group. With the help of this rotation group, the white noise analysis has explored new areas of mathematics and has extended the fields of applications.

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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
Table of Contents provided by Ingram. All Rights Reserved.

Lectures On White Noise Functionals

At a Glance

Hardcover

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