White Noise, 1994 An Infinite Dimensional Calculus Mathematics and Its Applications Series, Vol. 253

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.

1. Gaussian Spaces.- 2. J and f Transformation and the Decomposition Theorem.- 3. Generalized Functionals.- 4. The Spaces (f) and (f)*.- 5. Calculus of Differential Operators.- 6. Laplacian Operators.- 7. The Spaces D and D*.- 8. Stochastic Integration.- 9. Fourier and Fourier-Mehler Transforms.- 10. Dirichlet Forms.- 11. Applications to Quantum Field Theory.- 12. Feynman Integrals.- Appendices.- A.1 Hermite Polynomials.- A.2 Fock Space.- A.3 Reproducing Kernel Hilbert Spaces.- Notations and Conventions.