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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.

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Correspondence to Peter Mörters.

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Mörters, P. Almost sure Kallianpur–Robbins laws for Brownian motion in the plane. Probab Theory Relat Fields 118, 49–64 (2000). https://doi.org/10.1007/s004400000077

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  • DOI: https://doi.org/10.1007/s004400000077

Mathematics Subject Classification (2000)

  • Primary: 60J65, 60F65, 60J55

Key words and phrases

  • Planar Brownian motion
  • Additive functional
  • Almost sure limit theorems
  • Kallianpur-Robbins law
  • Ratio ergodic theorem
  • Chacon-Ornstein Theorem
  • Log averaging methods
  • Higher order logarithmic density