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#### Cut points on Brownian paths

(Institute of Mathematical Statistics, 1989-07)

Let X be a standard two-dimensional Brownian motion. There exists a.s. t [is an element of the set] (0; 1) such that X([0; t))[intersected with] X((t; 1]) = [empty set]. It follows that X([0; 1]) is not
homeomorphic to the Sierpinski carpet a.s.

#### Curvature of the convex hull of planar Brownian motion near its minimum point

(North-Holland (Elsevier), 1989-10)

Let f be a (random) real-valued function whose graph represents the boundary of the convex hull of planar Brownian motion run until time 1 near its lowest point in a coordinate system so that f is non-negative and f(0) = 0. The ratio of f(x) and |x|/|log |x|| oscillates near 0 between 0 and infinity a.s.

#### On Brownian Excursions in Lipschitz Domains. Part II: Local Asymptotic Distributions

(Birkhäuser Boston, Inc., 1989)

In this paper, we continue the study initiated in Burdzy and Williams (1986) of the local properties of Brownian excursions in Lipschitz domains.
The focus in part I was on local path properties of such excursions. In particular, a necessary and sufficient condition was given for Brownian excursions in a Lipschitz domain to ...

#### Geometric Properties of 2-dimensional Brownian Paths

(Springer-Verlag GmbH, 1989)

Let A be the set of all points of the plane C, visited by two-dimensional Brownian motion before time 1. With probability 1, all points of A are "twist points" except a set of harmonic measure zero. "Twist points" may be continuously approached in [the set that contains all those elements of complex numbers that are not in] ...