Luigi Accardi

Si Si

Noboru Watanabe

This volume aims to return to the starting point of the fields of infinite dimensional analysis and quantum probability, fields that are growing rapidly at present, and to seriously attempt mutual interaction between the two, with a view to enumerating and solving the many fundamental problems they entail. For such a purpose, we look for interdisciplinary bridges in mathematics including classical probability and to different branches of physics, in particular, research for new paradigms for information science on the basis of quantum theory.

Infinite Dimensional Analysis, Quantum Probability And Related Topics, Qp38 - Proceedings Of The International Conference

T Hida

' 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. Contents:Generalized White Noise FunctionalsElemental Random Variables and Gaussian ProcessesLinear Processes and Linear FieldsHarmonic Analysis Arising from Infinite Dimensional Rotation GroupComplex White Noise and Infinite Dimensional Unitary GroupCharacterization of Poisson NoiseInnovation TheoryVariational Calculus for Random Fields and Operator FieldsFour Notable Roads to Quantum Dynamics Readership: Researchers in probability and statistics and mathematical physics. Keywords:White Noise;Poisson Noise;Stochastic Analysis;Innovation;Random Field and Its Variation;Random Complexity;Infinite Dimensional Rotation GroupKey Features:A tightly edited distillation of a quarter of century of work by many scientistsContains the original ideas with some interpretation of development to recent worksCommentaries on each of the topics coveredReviews:“This monograph is a valuable guide to the world of white noise analysis originated by Hida. It is full of ideas, essentials and philosophy about why analysis based upon the white noise is needed, what the white noise functionals are, how to handle stochastic processes on an applicational basis to quantum physics, how to find a natural way to formulate the infinite-dimensional harmonic analysis, etc. This book is recommendable not only to ambitious graduate students in probability or applied analysis in a broad sense, but also to researchers who work on infinite-dimensional analysis or mathematical physics, as well as to any readers who are interested in white noise analysis.”Mathematical Reviews '

Lectures on White Noise Functionals

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.

Introduction to Hida Distributions

Takeyuki Hida

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.

L Accardi

H H Kuo

L Streit

N Obata

K Saitô

The topics discussed in this book can be classified into three parts: (i) Gaussian processes. The most general and in fact final representation theory of Gaussian processes is included in this book. This theory is still referred to often and its developments are discussed. (ii) White noise analysis. This book includes the notes of the series of lectures delivered in 1975 at Carleton University in Ottawa. They describe the very original idea of introducing the notion of generalized Brownian functionals (nowadays called “generalized white noise functionals”, and sometimes “Hida distribution”. (iii) Variational calculus for random fields. This topic will certainly represent one of the driving research lines for probability theory in the next century, as can be seen from several papers in this volume. Contents:General Theory of White Noise FunctionalsGaussian and Other ProcessesInfinite Dimensional Harmonic Analysis and Rotation GroupQuantum TheoryFeynman Integrals and Random FieldsVariational Calculus and Random FieldsApplication to Biology Readership: Graduate students and researchers in the fields of probability theory, functional analysis, statistics and theoretical physics. Keywords:White Noise;Gaussian;Brownian Motion;LÃ©vy Process;Canonical Representation;Stochastic Infinitesimal Equation;Generalized Functional;Innovation;Multiple Markov;Random FieldReviews:“This collection of papers is a tribute to one of the great researchers within stochastic analysis, Takeyuki Hida … An interesting appendix, however, is the collection of remarks at the end of the book … These remarks serve to put the various papers into perspective, and represent a valuable contribution.”Mathematical Reviews

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