In this chapter we study multiple points of d-dimensional Brownian motion. We shall see, for example, in which dimensions the Brownian path has double points and explore how many double points there are. This chapter also contains some of the highlights of the book: a proof that planar Brownian motion has points of infinite multiplicity, the intersection equivalence of Brownian motion and percolation limit sets, and the surprising dimension-doubling theorem of Kaufman.

Intersection of paths: Existence and Hausdorff dimension

Existence of intersections

Suppose that {B1 (t): t ≥ 0} and {B2(t): t ≥ 0} are two independent d-dimensional Brownian motions started in arbitrary points. The question we ask in this section is, in which dimensions the ranges, or paths, of the two motions have a nontrivial intersection, in other words whether there exist times t1, t1 > 0 such that B1(t1) = B2(t2). As this question is easy if d = 1 we assume d ≥ 2 throughout this section.

We have developed the tools to decide this question in Chapter 4 and Chapter 8. Keeping the path {B1(t): t ≥ 0} fixed, we have to decide whether it is a polar set for the second Brownian motion. By Kakutani's theorem, Theorem 8.20, this question depends on its capacity with respect to the potential kernel. As the capacity is again related to Hausdorff measure and dimension, the results of Chapter 4 are crucial in the proof of the following result.