This chapter is devoted to a few facts, several of which I learned from an interesting set of notes by Bryc [3], about and related to what probabilists call a characteristic function.

This chapter deals with some of the properties of Gaussian measures and the construction of families of Gaussian random variables.

This chapter is a continuation of the preceding one. Here we will derive a few more general properies of abstract Wiener spaces and then contstruct examples of what physicists call free Euclidean fields.

The theories of Gaussian measures and Hilbert spaces are inextricably related, and the goal of this and the next chapter is to explain and explore that relationship.

Gaussian measures and Gaussian processes provide the context for most of this concise textbook, appropriate for a single semester special topics course.

In this concluding chapter I will deal with several matters that are of a rather abstract nature. The first of these is the famous theorem of Radon and Nikodym, which can be viewed as a generalization of the results in § 3.3 and plays an important role in the proof of the Hahn decomposition theorem, which is an abstraction of the results in § 1.2.2 . The second is the abstraction of Lebesgue’s integration theory that results from thinking about integrals as linear functions.

‘A Concise Introduction to the Theory of Integration’ was once a best-selling Birkhäuser title which published 3 editions. This manuscript is a substantial revision of the material. Chapter one now in

In § 2.2 we used Itô’s idea to construct a coupling of the measures $$\mu _{t, n}$$ in ( 1.2.12 ). As most readers will have realized, our treatment would have been less cumbersome if we had a notion of integration that allowed me to replace the prescription in ( 2.2.1 ) by $$\begin{aligned} X_n(t,\mathbf {x})=\mathbf {x}+\int _0^t\sigma \bigl (X_n(\lfloor \tau \rfloor _n,\mathbf {x})\bigr )\, dw(\tau )+\int _0^tb\bigl (X_n(\lfloor \tau \rfloor _n,\mathbf {x})\bigr )\, d\tau \end{aligned}$$ and allowed me to show that such an expression converges to an expression like $$\begin{aligned} X(t,\mathbf {x})=\mathbf {x}+\int _0^t\sigma \bigl (X(\tau ,\mathbf {x})\bigr )\, dw(\tau )+\int _0^tb\bigl (X(\tau ,\mathbf {x})\bigr )\, d\tau . \end{aligned}$$

WIENER SPACE, REVISITED

A stochastic process is a parameterized family of random variables. For the most part, in this book the stochastic processes with which we will deal are parameterized by $$t\in [0,\infty )$$ , to be thought of as “time,” and take values in some Euclidean space $${\mathbb {R}^N}$$ . When attempting to analyze such a family $$\{X(t): t\ge 0\}$$ of random variables on a probability space $$(\varOmega ,\mathcal {F},{\mathbb {P}})$$ , the first step is to understand the distribution $$\mu _t$$ of X(t) for each $$t\ge 0$$ . Thus one is forced to consider paths $$t\rightsquigarrow \mu _t$$ in the space $$\mathbf {M}_1({\mathbb {R}^N})$$ of probability measures on $${\mathbb {R}^N}$$ , and, as usual, this leads one to consider the “tangent field” along the path. With this in mind, the goal of this chapter is to first make precise exactly what the tangent to a $$\mathbf {M}_1({\mathbb {R}^N})$$ -valued path is and to then develop a procedure for recovering the path from its tangent field.

Motivated by a conjecture about time-frequency translations of functions, several properties of the Bargmann-Fock space H and the Segal-Bargmann transform I are investigated in this note. In particular, a characterization is given of those square integrable functions phi on R such that z is an element of C bar right arrow I phi(z + zeta) is an element of C is in H for all zeta is an element of C.

This note has two goals. First, for those who have heard the term but do not know what it means, it provides a gentle introduction to Malliavin's calculus as it applies to degenerate parabolic partial differential equations. Second, it applies that theory to generalizations of Kolmogorov's example of a highly degenerate operator which is nonetheless hypoelliptic.