Electronic ISBN:  9781470402464 
Product Code:  MEMO/137/657.E 
List Price:  $49.00 
MAA Member Price:  $44.10 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 137; 1999; 95 ppMSC: Primary 60;
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 twodimensional 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.ReadershipGraduate students and research mathematicians working in probability.

Table of Contents

Chapters

0. Introduction

1. Preliminaries

2. Decomposition of Bessel processes

3. Random walk estimates

4. Estimates for approximate points of increase

5. Two and three angle estimates

6. The main estimate

7. Estimates for wedges

8. Filling in the gaps

9. Further results and problems


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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 twodimensional 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.

Chapters

0. Introduction

1. Preliminaries

2. Decomposition of Bessel processes

3. Random walk estimates

4. Estimates for approximate points of increase

5. Two and three angle estimates

6. The main estimate

7. Estimates for wedges

8. Filling in the gaps

9. Further results and problems