The theory of Gaussian Hilbert spaces developed in this book has strong connections to stochastic integration, in particular to Itô integrals with respect to Brownian motion. We treat these Itô integrals in the first section, and some extensions and related results in the following ones: stochastic integrals over general measure spaces in Section 2, the Skorohod integral in Section 3, and complex stochastic integrals and measures in Section 4.

Our treatment is self-contained, and we do not require that the reader has any prior knowledge of stochastic integration. On the other hand, such a knowledge would certainly be useful; we treat only those parts of stochastic integration theory that are directly relevant to the subject of this book, and many important topics are not included. For example, we consider only stochastic integrals with respect to Gaussian processes. Moreover, even for Brownian motion we do not include the fundamental Itô formula.

Hence, this chapter will perhaps be best understood in connection and comparison with other, more direct and complete, treatments of stochastic integration; see for example McKean (1969) and Protter (1990).

Brownian motion and Itô integrals

In this section, we assume that Bt, 0 ≤ t < ∞, is a standard Brownian motion and consider, as in Example 1.10, the Gaussian Hilbert space H = H(B) spanned by {Bt}t≥0.

The Malliavin calculus (also known as stochastic calculus of variation) is a differential calculus for functions (i.e. random variables) defined on a space with a Gaussian measure. (In applications, the space is usually some version of the Wiener space.) In accordance with our general principle, we present here a version concentrating on the random variables without explicit mention of the underlying space.

We define in Sections 1–3 the basic derivative operators ∂ξ and ▽ for an arbitrary Gaussian Hilbert space, and in Section 9 the dual divergence operator. We also give a detailed treatment of the Sobolev spaces Dk, p in Sections 5–8; this includes a proof of the important Meyer inequalities in Section 8. Results on existence and smoothness of densities are given in Sections 4 and 10; these results are central in many applications. Finally, a connection with the Skorohod integral is established in Section 11.

The first application of Malliavin calculus (Malliavin 1978) was to study smoothness of solutions to partial differential operators. Many other applications have been developed later, for example to stochastic differential equations and stochastic integrals. We will not treat any of these applications here; for applications, other versions of the theory and further results on analysis on Wiener space we refer to for example Bell (1987), Bouleau and Hirsch (1991), Ikeda and Watanabe (1984), Malliavin (1993, 1997), Nualart (1995, 1997+), Nualart and Zakai (1986), Ocone (1987), Peters (1997+), Stroock (1981), Üstünel (1995), Watanabe (1984), Zakai (1985).

In this chapter we assume that H is a Gaussian Hilbert space. We will use H to define and study several operators on L2(Ω, F(H), P). These operators are important in quantum physics, but we will not go into any such applications here; cf. for example Segal (1956), Glimm and Jaffe (1981, Chapter 6) and Meyer (1993).

There are also other applications of these operators, and we will use some of the results below in Chapter 15.

Some of the operators will be studied again in Chapters 15 and 16, where we also consider actions on Lp for p ≠ 2.

The reader may note that the operators treated here can be defined for any abstract Fock space based on an arbitrary Hilbert space, and that many of the results make sense in this generality, cf. for example Baez, Segal and Zhou (1992), Meyer (1993) and Parthasaraty (1992). Nevertheless, we will exclusively consider the Gaussian case here, where the extra structure is both helpful and, we hope, illuminating. (Of course, any result that can be stated for an abstract Fock space is valid in general as long as it is valid for the concrete realization treated here for Gaussian spaces.)

We warn the reader that several of the operators defined below will be unbounded and defined only on a dense subset of L2.