Brownian Motion

In this chapter we present some of the basic concepts from probability theory necessary for the main part of this book. No attempt has been made at either generality or completeness. Those concepts which provide motivation, or which are basic to our approach, are illustrated to some extent, whilst others will only be touched upon briefly. For example, certain specific properties of an infinite-dimensional probability measure (§1.3, (iii)) are discussed in some detail, as are some characterisations of Gaussian systems of random variables. Many theorems and propositions whose proofs can be found readily in standard texts will be stated without proof, or with only an outline of the proof. For further details of these, as well as related topics, the reader is referred to such books as K.ltô (1953c), W. Feller (1968,1971), and J. L. Doob (1953).

We begin this chapter with the definition of Brownian motion and a proof that its distribution is supported by the space of continuous functions (§2.1), and then go on to deal with important aspects of Brownian motion such as its sample path (§2.2) and Markov properties (§2.4). It is through these discussions that we can appreciate the place of Brownian motion within the class of all stochastic processes and, in particular, Gaussian processes. Two methods of constructing Brownian motion will be presented (§2.3), each of which is significant in its own right, and which also exhibits the ideas underlying constructions relevant to later chapters. Markov properties will only be touched upon briefly (§§2.4–2.6), but, hopefully, enough to enable a close connection with analysis to be seen.

As was announced in §1.3, (ii), this chapter is devoted to a study of the distributions of generalised stochastic processes. In the finite-dimensional case Bochner’s theorem establishes a one-to-one correspondence between distributions and characteristic functions, and this result has a counterpart in the present context, where the Bochner-Minlos theorem (§3.2) links probability measures on the space of generalised functions with characteristic functionals on the space of test functions. A secondary aim of this chapter is to provide a discussion of the generalised stochastic process white noise from several points of view (§§3.3, 3.4), this serving as background to much of what will be considered in the next chapter.

We are going to discuss those functionals, in general non-linear, of Brownian motion, which can be expressed in the form: .

Once again we begin with the triple E ⊆ L²(R) ⊆ E* where as in §3.1, E is a real nuclear space with dual space E* and the two spaces are linked by the canonical bilinear form <x, ξ >, x ∊ E*, ξ ∊ E.

Complex Gaussian systems are the most important families of complex-valued random variables, and this chapter begins by presenting the general background to such systems. We then observe that complex white noise, the white noise of Chapter 3 complexified, is a complex Gaussian system. Functionals of complex white noise may also be viewed as functionals of complex Brownian motion and the analysis of such functionals is not only useful in the study of stochastic processes, but is also widely used in applications. Consequently it is an important problem to express these in a concrete form and to develop ways of analysing them (§§6.2–6.3). On the other hand, the infinite dimensional unitary group arises naturally in the study of the probability measure determined on the complex-valued (generalised) function space by complex white noise. This unitary group plays the same role here as the infinite-dimensional rotation group did in describing properties of white noise (§7.1). For added interest, this group is intimately related to aspects of the theory of differential equations and quantum mechanics (§7.5–7.6).

As in the last chapter E denotes a nuclear space which is dense in L²(R). We write E _c and E _c ^* for the complexification of E and the dual space of E _c , respectively, and let be complex white noise.