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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$46
Individual Members: US$27.60
Institutional Members: US$36.80
Order Code: MEMO/137/657
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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.

Readership

Graduate students and research mathematicians working in probability.

Table of Contents

  • 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
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