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{1176991253,
author = {Pitman, Jim and Yor, Marc},
title = {Further Asymptotic Laws of Planar Brownian Motion},
journal = {Ann. Probab.},
volume = {17},
number = {4},
year = {1989},
pages = { 965-1011},
language = {en},
url = {http://dml.mathdoc.fr/item/1176991253}
}
Pitman, Jim; Yor, Marc. Further Asymptotic Laws of Planar Brownian Motion. Ann. Probab., Tome 17 (1989) no. 4, pp. 965-1011. http://gdmltest.u-ga.fr/item/1176991253/