## The Annals of Probability

### Further Asymptotic Laws of Planar Brownian Motion

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 17, Number 3 (1989), 965-1011.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991253

Digital Object Identifier
doi:10.1214/aop/1176991253

Mathematical Reviews number (MathSciNet)
MR1009441

Zentralblatt MATH identifier
0686.60085

JSTOR