This book provides the mathematical definition of white noise and gives its significance. White noise is in fact a typical class of idealized elemental (infinitesimal) random variables. Thus, we are naturally led to have functionals of such elemental random variables that is white noise. This book analyzes those functionals of white noise, particularly the generalized ones called Hida distributions, and highlights some interesting future directions. The main part of the book involves infinite dimensional differential and integral calculus based on the variable which is white noise.
The present book can be used as a supplementary book to Lectures on White Noise Functionals published in 2008, with detailed background provided.
Sample Chapter(s)
Chapter 1: Preliminaries and Discrete Parameter White Noise (481 KB)
Contents:
- Preliminaries and Discrete Parameter White Noise
- Continuous Parameter White Noise
- White Noise Functionals
- White Noise Analysis
- Stochastic Integral
- Gaussian and Poisson Noises
- Multiple Markov Properties of Generalized Gaussian Processes and Generalizations
- Classification of Noises
- Lévy Processes
Readership: Researchers in probability and statistics, in particular stochastic analysis.
“As a whole, the text is well written in a lucid style and is planned so nicely that the readers may grasp with ease the core part of white noise analysis and may also have a clear panoramic view of various aspects of its application, depending on their interest. This book is recommendable for potentially interested people or ambitious graduate students who are unfamiliar with generalized stochastic processes and infinite dimensional analysis. The prerequisite is only the basis of functional analysis and probability theory, and a little knowledge of stochastic processes and quantum theory are preferable. It is also suitable for physicists, applied physicists and physical engineers who are interested in applications to quantum physics, in particular, to Feynman path integrals. In the appendix, plenty of important formulae frequently appearing in white noise analysis are compactly summarized.”
Zentralblatt MATH
“In general, the book may be interesting to young researchers who need to obtain a quick introductory overview of white noise analysis. Potential readers should be familiar with basic probability theory and functional analysis. Overall, the book provides a good survey in a compact form with simple explanations and simplified proofs.”
Mathematical Reviews
