Open Access
July, 1989 Further Asymptotic Laws of Planar Brownian Motion
Jim Pitman, Marc Yor
Ann. Probab. 17(3): 965-1011 (July, 1989). DOI: 10.1214/aop/1176991253


The asymptotic distributions for large times of a variety of additive functionals of planar Brownian motion $Z$ are derived. Associated with each point in the plane, and with the point infinity, there is a complex Brownian motion governing the asymptotic behavior of windings of $Z$ close to that point. An independent Gaussian field over the plane governs fluctuations in local occupation times of $Z$, while a further independent family of complex Brownian sheets governs finer features of the windings of $Z$. These results unify and extend earlier results of Kallianpur and Robbins, Spitzer, Kasahara and Kotani, Messulam and the authors.


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Jim Pitman. Marc Yor. "Further Asymptotic Laws of Planar Brownian Motion." Ann. Probab. 17 (3) 965 - 1011, July, 1989.


Published: July, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0686.60085
MathSciNet: MR1009441
Digital Object Identifier: 10.1214/aop/1176991253

Primary: 60G65
Secondary: 60F05 , 60F17 , 60G44 , 60G55 , 60H05

Keywords: Additive functionals , Asymptotic distributions , Brownian motion , Brownian sheets , continuous martingales , singular integrals , Winding numbers

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 3 • July, 1989
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