Almost sure Kallianpur–Robbins laws for Brownian motion in the plane

Abstract

The Kallianpur–Robbins law describes the long term asymptotic behaviour of integrable additive functionals of Brownian motion in the plane. In this paper we prove an almost sure version of this result. It turns out that, differently from many known results, this requires an iterated logarithmic average. A similar result is obtained for the small scales asymptotic by means of an ergodic theorem of Chacon–Ornstein type, which allows an exceptional set of scales.