J and f Transformation and the Decomposition Theorem

Abstract

In this section we analyze further the structure of (L²) = L²(N*,B,μ). We shall establish the well-known Wiener—Itô decomposition theorem which states that (L²) has a direct sum decomposition into homogeneous chaos’ (see below). This theorem is due to Wiener (1938) and Itô (1951) in the case of the Wiener space (or the white noise space). In its general form this result has been proved first by Segal (1956). We refer the reader also to Hida (1980a) and Simon (1974). We shall prove the theorem here with the help of two important transformations, denoted by J and j, which will be introduced first. At the end of this chapter we shall specialize to the white noise space. In this case we shall provide the very powerful representation of elements in (L²) in terms of Wick powers of distributions. The ideas are closely related to “folklore wisdom” in quantum physics (e.g., Glimm and Jaffe (1981), Simon (1974)). The presentation of some parts of this chapter owes much to the article by Nelson (1973c).