Section 7.1 provides a brief introduction to the theory of martingales with a continuous parameter. As anyone at all familiar with the topic knows, anything approaching a full account of this theory requires much more space than a book like this can provide. Thus, I deal with only its most rudimentary aspects, which, fortunately, are sufficient for the applications to Brownian motion that I have in mind. Namely, in §7.2 I first discuss the intimate relationship between continuous martingales and Brownian motion (Lévy’s martingale characterization of Brownian motion), then derive the simplest (and perhaps most widely applied) case of the Doob–Meyer Decomposition Theory, and finally show what Burkholder’s Inequality looks like for continuous martingales. In the concluding section, §7.3, the results in §7.1 and §7.2 are applied to derive the Reflection Principle for Brownian motion.

This chapter is devoted to the study of infinitely divisible laws. It begins in §3.1 with a few refinements (especially the Lévy Continuity Theorem) of the Fourier techniques introduced in §2.3. These play a role in §3.2, where the Lévy–Khinchine formula is first derived and then applied to the analysis of stable laws.

Because they are not needed earlier, conditional expectations do not appear until Chapter 5. The advantage gained by this postponement is that, by the time I introduce them, I have an ample supply of examples to which conditioning can be applied; the disadvantage is that, with considerable justice, many probabilists feel that one is not doing probability theory until one is conditioning. Be that as it may, Kolmogorov’s definition is given in §5.1 and is shown to extend naturally to both σ-finite measure spaces and random variables with values in a Banach space. Section 5.2 presents Doob’s basic theory of real-valued, discrete parameter martingales: Doob’s Inequality, his Stopping Time Theorem, and his Martingale Convergence Theorem. In the last part of §5.2, I introduce reversed martingales and apply them to DeFinetti’s theory of exchangeable random variables.

The central topic here is the abstract theory of weak convergence of probability measures on a Polish space. The basic theory is developed in §9.1. In §9.2 I apply the theory to prove the existence of regular conditional probability distributions, and in §9.3 I use it to derive Donsker’s Invariance Principle (i.e., the pathspace statement of the Central Limit Theorem).

Chapter 1 contains a sampling of the standard, point-wise convergence theorems dealing with partial sums of independent random variables. These include the Weak and Strong Laws of Large Numbers as well as Hartman–Wintner’s Law of the Iterated Logarithm. In preparation for the law of the iterated logarithm, Cramér’s theory of large deviations from the law of large numbers is developed in §1.3. Everything here is very standard, although I feel that my passage from the bounded to the general case of the law of the iterated logarithm has been considerably smoothed by the ideas that I learned in conversation with M. Ledoux.

Chapter 2 is devoted to the classical Central Limit Theorem. The initial presentation is based on Lindeberg’s non-Fourier techniques. This is followed by a derivation of the Berry–Esseen estimate based on ideas of C. Stein. Fourier techniques are introduced in §2.3, and in the final section the CLT is used to derive W. Beckner’s sharp Lpestimates for the Fourier transform.

Chapter 8 provides an introduction to Gaussian measures on a Banach space from the point of view that originated in the work of N. Wiener and was further developed by L. Gross and I. Segal. The underlying idea is that, even though it cannot fit there, the measure would like to live on the Hilbert space (the Cameron–Martin space) for which it would be the standard Gauss measure, and it is in that Hilbert space that its properties are encoded. A good deal of functional analysis is required to carry out this program, and the estimate that makes the program possible is X. Fernique’s remarkable exponential estimate. Included are derivations of M. Schilder’s large deviations theorem for Brownian motion and V. Strassen’s function space version of the law of the iterated logarithm, both of which confirm the importance of the Cameron–Martin space.

In Chapter 4 I construct the Lévy processes (a.k.a. independent increment processes) corresponding to infinitely divisible laws. Section 4.1 provides the requisite information about the pathspace D(ℝN) of right-continuous paths with left limits, and §4.2 gives the construction of Lévy processes with discontinuous paths, the ones corresponding to infinitely divisible laws having no Gaussian part. Finally, in §4.3 I construct Brownian motion, the Lévy process with continuous paths, following the prescription given by Lévy. This section also contains a derivation of Kolmogorov’s continuity criterion for general Banach space-valued stochastic processes.

The intimate connection between Brownian motion of classical potential theory is described in Chapter 11. The first topic is again the representation of solutions to the Dirichlet problem in terms of the exit distribution of Brownian paths from a region. In particular, it is shown that, with probability 1, Brownian paths exit through regular points. This is followed by a discussion of the Poisson problem and its relationship, depending on dimension, to the transience or recurrence of Brownian paths. Among other things, a proof is given of F. Riesz’s representation theorem for superharmonic functions, and this result is used to introduce the concept of capacity. K. L. Chung’s formula for the capacitory potential in term of the last exit distribution of Brownian paths is derived and used to prove Wiener’s test for regularity in terms of capacity. Finally, the chapter concludes with two interesting connections, one made by F. Spitzer and the other by G. Hunt, between Brownian paths and capacity.

Chapter 10 is an introduction to the connections between probability theory and partial differential equations. At the beginning of §10.1, I show that martingales provide a link between probability theory and partial differential equations. More precisely, I show how to represent in terms of Wiener integrals solutions to parabolic and elliptic partial differential equations in which the Laplacian is the principal part. In the second part of §10.1, I derive the Feynman–Kac formula and use it to calculate various Wiener integrals. In §10.2 I introduce the Markov property of Wiener measure and show how it not only allows one to evaluate other Wiener integrals in terms of solutions to elliptic partial differential equations but also enables one to prove interesting facts about solutions to such equations as a consequence of their representation in terms of Wiener integrals. Continuing in the same spirit, I show in §10.2 how to represent solutions to the Dirichlet problem in terms of Wiener integrals, and in §10.3 I use Wiener measure to construct and discuss heat kernels related to the Laplacian and discuss ground states (a.k.a. stationary measures) for them.

Eigenvalues of the Laplacian matrix of a graph have been widely used in studying connectivity and expansion properties of networks, and also in analyzing random walks on a graph. Independently, statisticians introduced various optimality criteria in experimental design, the goal being to obtain more accurate estimates of quantities of interest in an experiment. It turns out that the most popular of these optimality criteria for block designs are determined by the Laplacian eigenvalues of the concurrence graph, or of the Levi graph, of the design. The most important optimality criteria, called A (average), D (determinant) and E (extreme), are related to the conductance of the graph as an electrical network, the number of spanning trees, and the isoperimetric properties of the graphs, respectively. The number of spanning trees is also an evaluation of the Tutte polynomial of the graph, and is the subject of the Merino–Welsh conjecture relating it to acyclic and totally cyclic orientations, of interest in their own right. This chapter ties these ideas together, building on the work in [4] and [5].