2

RICHARD F. BASS AND KRZYSZTOF BURDZY

Bertoin (1991) has a beautiful proof using a covering theorem. So far, there

is no two dimensional covering theorem which would give Theorem 0.1.

The proofs of Knight (1981) and Bertoin (1991) are very elegant applications

of powerful techniques - local time and coverings. Both proofs are short and based

on clever ways of looking at the problem. The proofs of Adelman (1985) and Burdzy

(1989) seem to be the most elementary of all proofs.

For any fixed unit vector v and d-dimensional Brownian motion Bt, the one

dimensional Brownian motion Bt • v does not have a point of increase, a.s., by

Theorem 0.2. Fubini's theorem easily implies that the set of unit vectors v for

which Bt - v has a point of increase has zero d — 1-dimensional Lebesgue measure

(as a subset of the unit sphere). A much stronger result is the following.

Theorem 0.3. Let Bt denote d-dimensional Brownian motion. The set oft 6 (0,1)

such that £([0, t)) and B((t, 1]) lie on opposite sides of some d — 1-dimensional

hyperplane has Hausdorff dimension zero.

Theorem 0.3 is well-known, but it seems never to have been published. It is

not hard to see that it can be proved using the methods of Evans (1985).

There exist results closely related to Theorem 0.1 which go in both "positive"

and "negative" directions. We start with a positive result.

Theorem 0.4. (Burdzy (1989)) Two-dimensional Brownian motion has cut points:

with probability 1, there exist times t such that Z([0,1] — {£}) is not a connected

set.

The original proof given in Burdzy (1989) contains a gap; see Burdzy (1995)

for the correction. A routine modification of the proof in Burdzy (1989) shows

that for each e, with probability p(e) 0, a Lipschitz curve can be found that

separates the two pieces Z([0,t)) and Z((t, 1]), such that the curve has Lipschitz

constant less than e. Lawler (1996) has a new proof of Theorem 0.4 and an estimate

for the Hausdorff dimension of cut points. The two-dimensional result easily im-

plies the existence of cut points for three dimensional Brownian motion. In higher

dimensions, all points on Brownian paths are cut points.

In order to state the next result, we need some notation. Let H(z,a) be

the half-plane obtained by first rotating the right half-plane by an angle a and

then translating it by the vector z. Let ^4(a1,(22) denote the event that for some

te (0,1), wehaveZ([0,£)) C H{Ztlal) and Z((t, 1]) c H(Zt,a2).

Theorem 0.5. (i) (Shimura (1988)) Fix any ai ^ a2. Then P(A(aua2)) = 0.

(ii) (Shimura (1992)) Fix any a 0. Then P(lJ-aa1,a2 a A(m + ^ 2 ) ) 0.

Theorem 0.5 (ii) is of special interest to us as it shows that Theorem 0.1 is a

very "sharp" result. This claim is reinforced by the following result, which asserts

the existence of cut planes in three and higher dimensions.