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We would like to thank him for allowing us to include his unpublished result in our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