We introduce a universal space for actions and equivalence relations via spaces of subshifts.

We survey the theory of free countable Borel equivalence relations.

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.

We survey the theory of amenable countable Borel equivalence relations.

We introduce essentially countable and reducible-to-countable Borel equivalence relations and discuss situations where they appear in various contexts.

We survey the role of structurability in the theory of countable Borel equivalence relations.

We introduce countable Borel equivalence relations and survey some of the fundamental structural results.

We introduce and study topological realizations of countable Borel equivalence relations.

We start the study of the structure of the Borel reducibility order on countable Borel equivalence relations.

We survey the theory of hyperfinite countable Borel equivalence relations.

This is a collection of many of the open problems discussed in the book.

We survey the theory of universal countable Borel equivalence relations.

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.