White Noise Distribution Theory
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Summary
Learn the basics of white noise theory with White Noise Distribution Theory. This book covers the mathematical foundation and key applications of white noise theory without requiring advanced knowledge in this area. This instructive text specifically focuses on relevant application topics such as integral kernel operators, Fourier transforms, Laplacian operators, white noise integration, Feynman integrals, and positive generalized functions. Extremely well-written by one of the field's leading researchers, White Noise Distribution Theory is destined to become the definitive introductory resource on this challenging topic.
Table of Contents
Introduction to White Noise
What is White Noise?
A Simple Example
Background
Abstract Wiener Spaces
Countably-Hilbert Spaces
Nuclear Spaces
Gel'fand Triples
White Noise as an Infinite Dimensional Calculus
White Noise Space
A Reconstruction of the Schwartz Space
The Space of Test and Generalized Functions
Some Examples of Test and Generalized Functions
Constructions of Test and Generalized Functions
General Ideas for Several Constructions
Construction from a Hilbert Space and an Operator
General Construction of Kubo and Takenaka
Construction of Kondratiev and Streit
The S-Transform
Wick Tensors and Multiple Weiner Integrals
Definition of the S-Transform
Examples of Generalized Functions
Continuous Versions and Analytic Extensions
Continuous Versions of Test Functions
Growth Condition and Norm Estimates
Analytic Extensions of Test Functions
Delta Functions
Donsker's Delta Function
Kubo-Yokoi Delta Function
Continuity of the Delta Functions
Characterization Theorems
Characterization of Generalized Functions
Convergence of Generalized Functions
Characterization of Test Functions
Wick Product and Convolution
Integrable Functions
Differential Operators
Differential Operators
Adjoint Operators
Multiplication Operators
Gross Differentiation and Gradient
Integral Kernel Operators
Heuristic Discussion
Integral Kernel Operators
Gross Laplacian and Number Operator
Lambda Operator
Translation Operators
Representation Theorem
Fourier Transforms
Definition of the Fourier Transform
Representations of the Fourier Transform
Basic Properties
Decomposition of the Fourier Transform
Fourier-Gauss Transforms
Characterization of the Fourier Transform
Fourier-Mehler Transforms
Initial Value Problems
Laplacian Operators
Semigroup for the Gross Laplacian
Semigroup for the Number Operator
Lévy Laplacian
Lévy Laplacian by the S-Transform
Spherical Mean and the Lévy Laplacian
Relationship Between Gross and Lévy Laplacians
Volterra Laplacian
Relationship with the Fourier Transform
Two-Dimensional Rotations
White Noise Integration
Informal Motivation
Pettis and Bochner Integrals
White Noise Integrals
An Extension of the Itô Integral
Generalization of Itô's Formula
One-Sided White Noise Differentiation
Stochastic Integral Equations
White Noise Integral Equations
Feynman Integrals
Informal Derivation
White Noise Formulation
Explicit Calculation
Positive Generalized Functions
Positive Generalized Functions
Construction of Lee
Characterization of Hida Measures
Appendix A: Notes and Comments
Appendix B: Miscellaneous Formulas
Bibliography
List of Symbols
Index
Editorial Reviews
"It is precisely written, up to date, and makes frequent appeal to the research literature."
-The Mathematical Gazette
