0. Introduction.
Let Zt denote two-dimensional Brownian motion and for a set A C [0, oo),
let Z(A) = {Zs : s e A}. If there exists a straight line C and a time t (0,1) such
that Zt 6 C, Z([0,t)) lies on one side of £, and Z((t, 1]) lies on the other side of £,
we say that £ is a cut line for Zt. Our main purpose is to present a proof of the
following result.
Theorem 0.1. Cut lines for 2-dimensional Brownian motion do not exist.
This answers Problem 8 in Taylor (1986). Prof. Taylor has informed us that
the problem is older than that.
A rigorous version of Theorem 0.1 is stated below as Theorem 0.7.
Taylor's problem has attracted much attention over the years and we will
first review some related results. We start with the well known one dimensional
counterpart of Theorem 0.1. Let Xt denote one dimensional Brownian motion.
Theorem 0.2. (Dvoretzky, Erdos and Kakutani (1961)) One dimensional Brown-
ian motion has no points of increase, a.s. In other words, with probability 1, there
is no t e (0,1) such that X([0, t)) H X((t, 1]) = 0.
The original proof of Theorem 0.2 is considered difficult, but a number of
alternative proofs have been found. It may be instructive to see why all attempts
at generalizing the proofs to higher dimensions have failed so far.
Knight (1981) and Berman (1983) (see also Karatzas and Shreve (1987))
proved the theorem using properties of Brownian local time. Local time is a function
of time and space variables. Under some assumptions, local time, viewed as a
function of the space variable, is a Markov process with known distribution and one
can prove that this Markov process does not visit 0. It is intuitively clear (although
it requires some argument) that the existence of a point of increase would imply
vanishing of the local time process in the space variable. The existence of local time
indexed by all straight lines in the plane for two dimensional Brownian motion has
been proved in Bass (1984). The distribution of this local time process, which could
conceivably be the key to a potential proof of Theorem 0.1, is elusive, though.
Short and elementary proofs of Theorem 0.2 were found by Adelman (1985)
and by Burdzy (1989). Aldous (1989) included in his book a heuristic argument
based on "Poisson clumping." Peres (1996) showed that one can first consider the
discrete analogue of the problem and then deduce Theorem 0.2 from its discrete
counterpart. Burdzy (1996) contains a proof based on a branching idea. Doney
(1996) has some related results on Levy processes. All of these proofs rely in
some way on the order structure of the real line and do not apply directly to two
dimensional processes.
Research partially supported by NSF grant DMS-9322689.
Received by the editor February 2, 1997, and in revised form September 18, 1997.
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