White Noise Distribution Theory: 1st Edition (Hardback) book cover

White Noise Distribution Theory

1st Edition

By Hui-Hsiung Kuo

CRC Press

400 pages

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Hardback: 9780849380778
pub: 1996-04-17
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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.


"It is precisely written, up to date, and makes frequent appeal to the research literature."

-The Mathematical Gazette

Table of Contents

Introduction to White Noise

What is White Noise?

A Simple Example


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


List of Symbols


About the Series

Probability and Stochastics Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General