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On occupation times of the first and third quadrants for planar Brownian motion

Part of: Markov processes

Published online by Cambridge University Press:  04 April 2017

Philip A. Ernst  and
Larry Shepp
Philip A. Ernst*
Affiliation:
Rice University
Larry Shepp*
Affiliation:
University of Pennsylvania
*
* Postal address: Department of Statistics, Rice University, 6100 Main Street, Houston, Texas, TX 77005, USA.
** 9 September 1936–23 April 2013
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Abstract

In Bingham and Doney (1988) the authors presented the applied probability community with a question which is very simply stated, yet is extremely difficult to solve: what is the distribution of the quadrant occupation time of planar Brownian motion? In this paper we study an alternate formulation of this long-standing open problem: let X(t), Y(t) t≥0, be standard Brownian motions starting at x, y, respectively. Find the distribution of the total time T=Leb{t∈[0,1]: X(t)×Y(t)>0}, when x=y=0, i.e. the occupation time of the union of the first and third quadrants. If two adjacent quadrants are used, the problem becomes much easier and the distribution of T follows the arcsine law.

Keywords

Planar Brownian motionoccupation timeKontorovich–Lebedev

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 337 - 342
Copyright
Copyright © Applied Probability Trust 2017 

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