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Cutting Brownian Paths
Richard F. Bass and Krzysztof Burdzy, University of Washington, Seattle, WA
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Memoirs of the American Mathematical Society
1999; 95 pp; softcover
Volume: 137
ISBN-10: 0-8218-0968-7
ISBN-13: 978-0-8218-0968-6
List Price: US$49 Individual Members: US$29.40
Institutional Members: US\$39.20
Order Code: MEMO/137/657

A long open problem in probability theory has been the following: Can the graph of planar Brownian motion be split by a straight line?

Let $$Z_t$$ be two-dimensional Brownian motion. Say that a straight line $$\mathcal L$$ is a cut line if there exists a time $$t \in (0,1)$$ such that the trace of $$\{ Z_s: 0 \leq s < t\}$$ lies on one side of $$\mathcal L$$ and the trace of $$\{Z_s: t < s < 1\}$$ lies on the other side of $$\mathcal L$$. In this volume, the authors provide a solution, discuss related works, and present a number of open problems.

Graduate students and research mathematicians working in probability.

• Introduction
• Preliminaries
• Decomposition of Bessel processes
• Random walk estimates
• Estimates for approximate points of increase
• Two and three angle estimates
• The main estimate
• Estimates for wedges
• Filling in the gaps
• Further results and problems
• References