An exploration of random fields. The authors use an approach that firstly constructs innovation, which is the most elemental stochastic process with a basic and simple way of dependence, and then they express the given field as a function of the innovation.

A random field is a mathematical model of evolutional fluctuating complex systems parametrized by a multi-dimensional manifold like a curve or a surface. As the parameter varies, the random field carries much information and hence it has complex stochastic structure.The authors of this book use an approach that is characteristic: namely, they first construct innovation, which is the most elemental stochastic process with a basic and simple way of dependence, and then express the given field as a function of the innovation. They therefore establish an infinite-dimensional stochastic calculus, in particular a stochastic variational calculus. The analysis of functions of the innovation is essentially infinite-dimensional. The authors use not only the theory of functional analysis, but also their new tools for the study.

A random field is a mathematical model of evolutional fluctuatingcomplex systems parametrized by a multi-dimensional manifold like acurve or a surface. As the parameter varies, the random field carriesmuch information and hence it has complex stochastic structure.The authors of this book use an approach that is characteristic: namely, they first construct innovation, which is the most elementalstochastic process with a basic and simple way of dependence, and thenexpress the given field as a function of the innovation. Theytherefore establish an infinite-dimensional stochastic calculus, inparticular a stochastic variational calculus. The analysis offunctions of the innovation is essentially infinite-dimensional. Theauthors use not only the theory of functional analysis, but also theirnew tools for the study

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.

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.

This volume includes papers by leading mathematicians in the fields of stochastic analysis, white noise theory and quantum information, together with their applications. The papers selected were presented at the International Conference on Stochastic Analysis: Classical and Quantum held at Meijo University, Nagoya, Japan from 1 to 5 November 2004. The large range of subjects covers the latest research in probability theory.

The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics, including probability theory, differential geometry, harmonic analysis, and special functions. The present volume unifies many results scattered throughout the mathematical, physical, and engineering literature, as well as it introduces new results from this area first proved by the author. The book also presents many practical applications, in particular in such highly interesting areas as approximation theory, cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes, statistics, functional analysis, astronomy, and engineering.

The main purpose of this volume is to emphasize the multidisciplinary aspects of this very active new line of research in which concrete technological and industrial realizations require the combined efforts of experimental and theoretical physicists, mathematicians and engineers. Contents: Coherent Quantum Control of -Atoms Through the Stochastic Limit (L Accardi); Information, Innovation and Elemental Random Field (T Hida); Joint Extension of States of Fermion Subsystems (H Araki); Emergence of White Noise Equations from Classical Quantum Mechanics (A Boukas); Saturation of an Entropy Bound and Quantum Markov States (D Petz); Quantum Entanglement, Purification, and Linear-Optics Quantum Gates with Photonic Qubits (P Walther & A Zeilinger); Group Theory of Dynamical Maps (E C G Sudarshan); Quantum Logical Gates Realized by Beam Splittings (W Freudenberg et al.); Generalized Sectors and Adjunctions (I Ojima); Note on Quantum Mutual Type Measures and Capacity (N Watanabe); Structure of Linear Processes (S Si); An Infinite Dimensional Laplacian Acting on Some Class of Levy White Noise Functionals (K Saito); Fidelity of Quantum Teleportation Model Using Beam Splittings (K-H Fichtner et al.); Noncanonical Representations of a Multi-Dimensional Brownian Motion (Y Hibino); and other papers. Readership: Researchers in quantum physics and theoretical physics.

This volume highlights recent developments of stochastic analysis with a wide spectrum of applications, including stochastic differential equations, stochastic geometry, and nonlinear partial differential equations.While modern stochastic analysis may appear to be an abstract mixture of classical analysis and probability theory, this book shows that, in fact, it can provide versatile tools useful in many areas of applied mathematics where the phenomena being described are random. The geometrical aspects of stochastic analysis, often regarded as the most promising for applications, are specially investigated by various contributors to the volume.

This volume takes up various topics in Mathematical Analysis including boundary and initial value problems for Partial Differential Equations and Functional Analytic methods.Topics include linear elliptic systems for composite material ? the coefficients may jump from domain to domain; Stochastic Analysis ? many applied problems involve evolution equations with random terms, leading to the use of stochastic analysis.The proceedings have been selected for coverage in: ? Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings)? CC Proceedings ? Engineering & Physical Sciences