## 60F15 Strong theorems

### Refine

#### Keywords

- Kallianpur-Robbins law (1)
- higher order (1)
- log averaging methods (1)
- occupation measure (1)
- planar Brownian motion (1)
- ratio ergodic theorem (1)
- strong theorems (1)

- Pathwise Kallianpur-Robbins laws for Brownian motion in the plane (1998)
- The Kallianpur-Robbins law describes the long term asymptotic behaviour of the distribution of the occupation measure of a Brownian motion in the plane. In this paper we show that this behaviour can be seen at every typical Brownian path by choosing either a random time or a random scale according to the logarithmic laws of order three. We also prove a ratio ergodic theorem for small scales outside an exceptional set of vanishing logarithmic density of order three.