Abstract. Let Zt be two-dimensional Brownian motion. We say that a straight line
£ is a cut line if there exists a time t G (0,1) such that the trace of {Zs : 0 s t]
lies on one side of C and the trace o f { Z
: t s l } lies on the other side of C. In
this paper we prove that with probability one cut lines do not exist. This provides
a solution to Problem 8 in Taylor (1986).
A M S Subject Classification (1990): Primary 60J65; Secondary 60G17
Ke y words: Planar Brownian motion, cut lines, cut points, exceptional sets, Tay-
lor's problem, Bessel processes, conditioned Brownian motion, cones, random walks,
wedges, points of increase, convex hull
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