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White Noise : An Infinite Dimensional Calculus - T. Hida

White Noise

An Infinite Dimensional Calculus

Hardcover

Published: 31st March 1993
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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.

Gaussian Spacesp. 1
[Actual symbol not reproducible] and [actual symbol not reproducible] Transformation and the Decomposition Theoremp. 10
Generalized Functionalsp. 35
The Spaces [actual symbol not reproducible]p. 74
Calculus of Differential Operatorsp. 146
Laplacian Operatorsp. 184
The Spaces D and D[superscript*]p. 232
Stochastic Integrationp. 277
Fourier and Fourier-Mehler Transformsp. 317
Dirichlet Formsp. 366
Application to Quantum Field Theoryp. 399
Feynman Integralsp. 435
App. A.1 Hermite Polynomialsp. 451
App. A.2 Fock Spacep. 456
App. A.3 Reproducing Kernel Hilbert Spacesp. 468
App. A.4 Frechet and Gateaux Derivatives on [actual symbol not reproducible]p. 472
App. A.5 Topologies on [actual symbol not reproducible]p. 477
Bibliographyp. 486
Notations and Conventionsp. 507
Indexp. 513
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792322337
ISBN-10: 0792322339
Series: MATHEMATICS AND ITS APPLICATIONS (KLUWER )
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 520
Published: 31st March 1993
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 25.4 x 17.15  x 3.18
Weight (kg): 0.92