The Wiener mathematical model of the phenomenon observed by an English botanist Robert Brown in 1828 has been and still is one of the most interesting stochastic processes. Kingman writes that the deepest results in the theory of random processes are concerned with the interplay of the two most fundamental processes: Brownian motion and the Poisson process. Revuz and Yor point out that the Wiener process “is a good topic to center a discussion around because Brownian motion is in the intersection of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with independent increments”. Moreover, it belongs to the important class of diffusion processes. It is actually quite hard to find a book on probability and stochastic processes that does not describe this process at least in a heuristic way. Not a serious book, anyway.

Historically, Brown noted that pollen grains suspended in water perform a continuous swarming motion. Years (almost a century) later Bachelier and Einstein derived the probability distribution of a position of a particle performing such a motion (the Gaussian distribution) and pointed out its Markovian nature – lack of memory, roughly speaking. But it took another giant, notably Wiener, to provide a rigorous mathematical construction of a process that would satisfy the postulates of Einstein and Bachelier.

It is hard to overestimate the importance of this process.

This book is an expanded version of lecture notes for the graduate course “An Introduction to Methods of Functional Analysis in Probability and Stochastic Processes” that I gave for students of the University of Houston, Rice University, and a few friends of mine in Fall, 2000 and Spring, 2001. It was quite an experience to teach this course, for its attendees consisted of, on the one hand, a group of students with a good background in functional analysis having limited knowledge of probability and, on the other hand, a group of statisticians without a functional analysis background. Therefore, in presenting the required notions from functional analysis, I had to be complete enough for the latter group while concise enough so that the former would not drop the course from boredom. Similarly, for the probability theory, I needed to start almost from scratch for the former group while presenting the material in a light that would be interesting for the latter group. This was fun. Incidentally, the students adjusted to this challenging situation much better than I.

In preparing these notes for publication, I made an effort to make the presentation self-contained and accessible to a wide circle of readers. I have added a number of exercises and disposed of some. I have also expanded some sections that I did not have time to cover in detail during the course. I believe the book in this form should serve first year graduate, or some advanced undergraduate students, well. It may be used for a two-semester course, or even a one-semester course if some background is taken for granted.

A characteristic of functional analysis is that it does not see functions, sequences, or measures as isolated objects but as elements or points in a space of functions, a space of sequences, or a space of measures. In a sense, for a functional analyst, particular properties of a certain probability measure are not important; rather, properties of the whole space or of a certain subspace of such measures are important. To prove existence or a property of an object or a group of objects, we would like to do it by examining general properties of the whole space, not by examining these objects separately. There is both beauty and power in this approach. We hope that this crucial point of view will become evident to the reader while he/she progresses through this chapter and through the whole book.

Linear spaces

The central notion of functional analysis is that of a Banach space. There are two components of this notion: algebraic and topological. The algebraic component describes the fact that elements of a Banach space may be meaningfully added together and multiplied by scalars. For example, given two random variables, X and Y, say, we may think of random variables X + Y and αX (and αY) where α ∈ ℝ. In a similar way, we may think of the sum of two measures and the product of a scalar and a measure. Abstract sets with such algebraic structure, introduced in more detail in this section, are known as linear spaces. The topological component of the notion of a Banach space will be discussed in Section 2.2.

Bibliographical notes

Notes to Chapter 1 Rudiments of measure theory may be found in. Classics in this field are and; see also. A short but excellent account on convex functions may be found in, Chapter V, Section 8. A classical detailed treatment may be found in. The proof of the Steinhaus Theorem is taken from.

Notes to Chapter 2 There are many excellent monographs devoted to Functional Analysis, including. Missing proofs of the statements concerning locally compact spaces made in 2.3.25 may be found in and.

Notes to Chapter 3 Among the best references on Hilbert spaces are and. The proof of Jensen's inequality is taken from; different proofs may be found in and. Some exercises in 3.3 were taken from and. An excellent and well-written introductory book on martingales is; the proof of the Central Limit Theorem is taken from this book. Theorems 3.6.5 and 3.6.7 are taken from. A different proof of 3.6.7 may be found e.g. in.

Notes to Chapter 4 Formula (4.11) is taken from. Our treatment of the Itô integral is largely based on. For detailed information on matters discussed in 4.4.8 see e.g., and. To be more specific: for integrals with respect to square integrable martingales see e.g. Proposition 3.4 p. 67, Corollary 5.4 p. 78, Proposition 6.1. p. 79, Corollary 5.4, and pp. 279–282 in, or Chapter 3 in or Chapter 2 in. See also, etc.