Ebook: White Noise: An Infinite Dimensional Calculus

Many areas of applied mathematics call for an efficient calculus in infinite dimensions. This is most apparent in quantum physics and in all disciplines of science which describe natural phenomena by equations involving stochasticity. With this monograph we intend to provide a framework for analysis in infinite dimensions which is flexible enough to be applicable in many areas, and which on the other hand is intuitive and efficient. Whether or not we achieved our aim must be left to the judgment of the reader. This book treats the theory and applications of analysis and functional analysis in infinite dimensions based on white noise. By white noise we mean the generalized Gaussian process which is (informally) given by the time derivative of the Wiener process, i.e., by the velocity of Brownian mdtion. Therefore, in essence we present analysis on a Gaussian space, and applications to various areas of sClence. Calculus, analysis, and functional analysis in infinite dimensions (or dimension-free formulations of these parts of classical mathematics) have a long history. Early examples can be found in the works of Dirichlet, Euler, Hamilton, Lagrange, and Riemann on variational problems. At the beginning of this century, Frechet, Gateaux and Volterra made essential contributions to the calculus of functions over infinite dimensional spaces. The important and inspiring work of Wiener and Levy followed during the first half of this century. Moreover, the articles and books of Wiener and Levy had a view towards probability theory.

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This monograph presents a framework for infinite dimensional analysis based on white noise. This approach, which has many areas of application is both intuitive and efficient.
Among the concepts and structures generalized to an infinite dimensional setting in this book are: spaces of test and generalized functions, differential calculus, Laplacian and Fourier transforms and Dirichlet forms and their Markov processes. A multitude of concepts, such as Brownian motion functionals, falls into this framework. This book presents a simple, yet general theory of stochastic integration and also discusses construction quantum field theory and Feynman's functional integration.
This volume will be of interest to mathematicians and scientists who use stochastic methods in their research. The book will be of particular value to mathematicians in probability theory, functional analysis, measure theory, potential theory, as well as to physicists and scientists in engineering.

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This monograph presents a framework for infinite dimensional analysis based on white noise. This approach, which has many areas of application is both intuitive and efficient.
Among the concepts and structures generalized to an infinite dimensional setting in this book are: spaces of test and generalized functions, differential calculus, Laplacian and Fourier transforms and Dirichlet forms and their Markov processes. A multitude of concepts, such as Brownian motion functionals, falls into this framework. This book presents a simple, yet general theory of stochastic integration and also discusses construction quantum field theory and Feynman's functional integration.
This volume will be of interest to mathematicians and scientists who use stochastic methods in their research. The book will be of particular value to mathematicians in probability theory, functional analysis, measure theory, potential theory, as well as to physicists and scientists in engineering.
Content:
Front Matter....Pages i-xiii
Gaussian Spaces....Pages 1-9
J and f Transformation and the Decomposition Theorem....Pages 10-34
Generalized Functionals....Pages 35-73
The Spaces (f) and (f)*....Pages 74-145
Calculus of Differential Operators....Pages 146-183
Laplacian Operators....Pages 184-231
The Spaces D and D*....Pages 232-276
Stochastic Integration....Pages 277-316
Fourier and Fourier-Mehler Transforms....Pages 317-365
Dirichlet Forms....Pages 366-398
Applications to Quantum Field Theory....Pages 399-434
Feynman Integrals....Pages 435-450
Back Matter....Pages 451-516

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This monograph presents a framework for infinite dimensional analysis based on white noise. This approach, which has many areas of application is both intuitive and efficient.
Among the concepts and structures generalized to an infinite dimensional setting in this book are: spaces of test and generalized functions, differential calculus, Laplacian and Fourier transforms and Dirichlet forms and their Markov processes. A multitude of concepts, such as Brownian motion functionals, falls into this framework. This book presents a simple, yet general theory of stochastic integration and also discusses construction quantum field theory and Feynman's functional integration.
This volume will be of interest to mathematicians and scientists who use stochastic methods in their research. The book will be of particular value to mathematicians in probability theory, functional analysis, measure theory, potential theory, as well as to physicists and scientists in engineering.
Content:
Front Matter....Pages i-xiii
Gaussian Spaces....Pages 1-9
J and f Transformation and the Decomposition Theorem....Pages 10-34
Generalized Functionals....Pages 35-73
The Spaces (f) and (f)*....Pages 74-145
Calculus of Differential Operators....Pages 146-183
Laplacian Operators....Pages 184-231
The Spaces D and D*....Pages 232-276
Stochastic Integration....Pages 277-316
Fourier and Fourier-Mehler Transforms....Pages 317-365
Dirichlet Forms....Pages 366-398
Applications to Quantum Field Theory....Pages 399-434
Feynman Integrals....Pages 435-450
Back Matter....Pages 451-516
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