Introduction

Summary

A Gaussian Hilbert space is a (complete) linear space of random variables with (centred) Gaussian distributions. This simple notion combines probability theory and Hilbert space theory into a rich and powerful structure, and Gaussian Hilbert spaces and connected notions such as the Wiener chaos decomposition and Wick products appear in several areas of probability theory and its applications, for example in stochastic processes and fields, stochastic integration, quantum field theory and limit theory for various statistics. There are also applications to non-probabilistic analysis, for example Banach space geometry and partial differential equations.

Although there are many references dealing with such applications where Gaussian spaces are treated and used, see for example Hida and Hitsuda (1976), Hida, Kuo, Potthoff and Streit (1993), Holden, Øksendal, Ubøe and Zhang (1996), Ibragimov and Rozanov (1970), Kahane (1985), Kuo (1996), Major (1981), Malliavin (1993, 1997), Meyer (1993), Neveu (1968), Nualart (1995, 1997+), Obata (1994), Pisier (1989), Simon (1974, 1979a), Watanabe (1984), there seems to be a shortage of works dealing with the basic properties of Gaussian spaces in general, without connecting them to a particular application. (One exception is the paper by Dobrushin and Minlos (1977).) This book is an attempt to fill the gap by providing a collection of the most important definitions and results for general Gaussian spaces, together with some applications to special Gaussian spaces.