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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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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