CUTTING BROWNIAN PATHS

3

Theorem 0.6. Let Bt denote d-dimensional Brownian motion, where d 3. With

positive probability, there exists t € (0,1) such that B([0,t)) and B((t, 1]) lie on

the opposite sides of some (d — 1)-dimensional hyperplane.

We will provide a sketch of the proof in Section 9. The argument uses

estimates which are similar to those derived in the proof of Theorem 0.1. We will

skip the technical details in the hope that the reader will be able to reconstruct

a rigorous proof from our sketch, using the arguments developed in the earlier

sections.

We learned Theorem 0.6 from Robin Pemantle (private communication).

We would like to thank him for allowing us to include his unpublished result in our

paper together with our own proof.

A version of Theorem 0.1 stated below asserts that "local cut lines" do not

exist. Let Zt = (Xt,Yt) be two dimensional Brownian motion, and for 9 € [0,27r),

set

Zt{0) = Xt cos O + YtsinO,

the component of Zt in the 0 direction. Let

S = {3t,ft 0,0 € [0,2TT) : Zs(0) Zt(0) for (t - h) V 0 s t, (0.1)

Zu(0) Zt(0) for t u t + h}.

Theorem 0.7. P(«S) = 0.

We give a sketch of our argument. Let us say that a 1-dimensional Brownian

motion starting at 0 has an approximate point of increase of size e at level x G

(0,1) if after hitting the level x, the Brownian motion reaches the level 2 before

returning to x — e. It is not hard to show (cf. Burdzy (1990)) that if we fix

a then the probability that the one dimensional Brownian motion Zt(a) has an

approximate point of increase is of order l/log(l/£). This immediately implies

that the expected number of directions of the form a = m/\og(l/e), m integer,

with an approximate point of increase is finite. This estimate, however, is not

good enough. We circumvent it by conditioning on having an approximate point

of increase in a fixed direction. We calculate the probabilities for the occurrence of

approximate points of increase in a second fixed direction and then for having such

points in two other fixed directions. A second moment argument then gives a better

bound for the probability of having an approximate point of increase in one of the

directions m/log(l/£). The argument is further complicated by the facts that the

second moment argument must be iterated to get a sufficiently good bound and that

we are only able to obtain the probabilities for an approximate point of increase in

two or more directions for certain levels. This conditional second moment argument

was used in a somewhat simpler setting in Bass and Burdzy (1996).

The first part of the proof shows that with high probability there are no

approximate points of increase at angles ra/log(l/£), m integer. The second part