Almost sure Kallianpur-Robbins laws for Brownian motion in the plane (Q1590679)

Let \((B_t)_{t\geq 0}\) denote a linear or planar Brownian motion and let \((A(t),\;t\geq 0)\) be an integrable additive functional of it with associated measure \(\nu_A\). Then the Kallianpur-Robbins law states that in the linear case the asymptotic distribution of \(A(t)/\sqrt t\) is that of the modulus of a normal random variable with mean zero and variance \(\|\nu_A \|^2\) whereas in the planar case the asymptotic distribution of \(A(t)/ \sqrt{\log t}\) is the exponential distribution with parameter \(\pi/ \|\nu_A\|.\) This paper deals with almost sure versions of these limit theorems using logarithmic averages, as e.g., \[ \lim_{t\to \infty} {1\over\log t}\int^t_0 f\left({A(s) \over \sqrt s}\right) {ds\over s} \overset \text{a.s.} =\sqrt{2 \over\pi} \int^\infty_0 f\bigl(s\|\nu_A\|\bigr) e^{-s^2/2}ds \] in the linear case, where \(f\) is a suitable function. One of the main results here is the related one for the planar Brownian motion where, however, a stronger mean (iterated logarithmic average) is needed. The proof uses a special functional \(A\) (occupation time) and then applies a ratio ergodic theorem. A similar result is given using small scales instead of large times, therefore a ratio ergodic theorem of Chacon-Ornstein type is provided.