Si Si. Introduction to Hida Distributions

Singapore: World Scientific Publishing Company, 2011. — 268 p.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.ThePreliminaries and Discrete Parameter White Noise
Preliminaries
Discrete parameter white noise
Invariance of the measure µ
Harmonic analysis arising from O(E) on the space of functionals of Y = {Y (n)}
Quadratic forms
Differential operators and related operators
Probability distributions and Bochner-Minlos theorem
Continuous Parameter White Noise
Gaussian system
Continuous parameter white noise
Characteristic functional and Bochner-Minlos theorem
Passage from discrete to continuous
Stationary generalized stochastic processes
White Noise Functionals
In line with standard analysis
White noise functionals
Infinite dimensional spaces spanned by generalized linear functionals of white noise
Some of the details of quadratic functionals of white noise
The T -transform and the S-transform
White noise (t) related to δ-function
Infinite dimensional space generated by Hermite polynomials in (t)’s of higher degree
Generalized white noise functionals
Approximation to Hida distributions
Renormalization in Hida distribution theory
White Noise Analysis
Operators acting on (L2)-
Application to stochastic differential equation
Differential calculus and Laplacian operators
Infinite dimensional rotation group O(E)
Addenda
Stochastic IntegralWiener integrals and multiple Wiener integrals
The Ito integral
Hitsuda-Skorokhod integrals
Levy’s stochastic integral
Addendum : Path integrals
Gaussian and Poisson Noises
Poisson noise and its probability distribution
Comparison between the Gaussian white noise and the Poisson noise, with the help of characterization of measures
Symmetric group in Poisson noise analysis
Spaces of quadratic Hida distributions and their dualities
Multiple Markov Properties of Generalized Gaussian Processes and Generalizations
A brief discussion on canonical representation theory for Gaussian processes and multiple Markov property
Duality for multiple Markov Gaussian processes in the restricted sense
Uniformly multiple Markov processes
The N-ple Markov property of a generalized Gaussian process
Representations of multiple Markov generalized Gaussian processes
Stationary N-ple Markov generalized Gaussian process
Remarks on the definition of multiple Markov property
Classification of Noises
The birth of noises
A noise of new type
Invariance
Representation of linear generalized functionals of P′(u)’s
Nonlinear functionals of P′ with special emphasis onquadratic functionals
Observations
Levy Processes
Additive process and Levy process
Compound Poisson process
Stable distribution
Appendix
Sobolev spaces
Hermite polynomials
Rademacher functions
Spectral decomposition of covariance function
Variational calculus for random fields