Almost sure Kallianpur–Robbins laws for Brownian motion in the plane |
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Authors: | Peter Mörters |
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Institution: | 1.Universit?t Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67663 Kaiserslautern, Germany. e-mail: peter@mathematik.uni-kl.de,Germany |
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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.
Received: 8 May 1998 / Revised version: 1 December 1999 / Published online: 8 August 2000 |
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