Asymptotic laws of planar Brownian motion. (English) Zbl 0607.60070

Pitman, Jim; Yor, Marc

Ann. Probab. 14, 733-779 (1986).

This paper presents a unified approach to various asymptotic results for planar Brownian motion, involving such objects as additive functionals or windings about points of the plane. In particular, the authors obtain the joint asymptotic law of the windings about several points of the plane, thus extending a result due to F. Spitzer [Trans. Am. Math. Soc. 87, 187-197 (1958; Zbl 0089.136)] in the case of a single point.
The main tools are stochastic calculus, especially Knight’s theorem on orthogonal continuous martingales, and excursion theory. A very important role is also played by the representation of planar Brownian motion in polar coordinates (the so-called skew-product representation), which is used simultaneously with several different origins.

Reviewer: J.F.Le Gall

MSC:

60J65	Brownian motion
60J55	Local time and additive functionals
60F05	Central limit and other weak theorems
60H05	Stochastic integrals
60G44	Martingales with continuous parameter

Keywords:

planar Brownian motion; additive functionals; windings about points of the plane; Knight’s theorem

Citations:

Zbl 0089.136