Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-12T01:33:41.989Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bateman, H. and Erdélyi, A. (ed.) (1954).Tables of Integral Transforms, Vol. I.McGraw-Hill,New York.Google Scholar
[2] Bateman, H. and Erdélyi, A. (ed.) (1954).Tables of Integral Transforms, Vol. II.McGraw-Hill,New York.Google Scholar
[3] Bingham, N. H. and Doney, R. A. (1988).On higher-dimensional analogues of the arc-sine law.J. Appl. Prob. 25,120–131.CrossRefGoogle Scholar
[4] Desbois, J. (2007).Occupation times for planar and higher dimensional Brownian motion.J. Phys. A 40,2251–2262.Google Scholar
[5] Karatzas, I. and Shreve, S. E. (1991).Brownian Motion and Stochastic Calculus, 2nd edn (Grad. Texts Math. 113).Springer,New York.Google Scholar
[6] Kontorovich, M. J. and Lebedev, N. N. (1938).One method for solving certain problems of diffraction theory and the problems related to it.J. Exper. Theoret. Phys. USSR 8,1192–1206.Google Scholar
[7] Kontorovich, M. J. and Lebedev, N. N. (1939).On a method of solution of some problems of the diffraction theory.Acad. Sci. USSR J. Phys. 1,229–241.Google Scholar
[8] Lévy, P. (1965).Processus Stochastiques et Mouvement Brownien, 2nd edn.Gauthier-Villars,Paris.Google Scholar
[9] Oberhettinger, F. and Badii, L. (1973).Tables of Laplace Transforms.Springer,New York.CrossRefGoogle Scholar
[10] Pemantle, R., Peres, Y., Pitman, J. and Yor, M. (2001).Where did the Brownian particle go?.Electron. J. Prob. 6, 22pp.Google Scholar
[11] Pitman, J. (1996).Cyclically stationary Brownian local time processes.Prob. Theory Relat. Fields 106,299–329.Google Scholar