×

The packing measure of planar Brownian motion. (English) Zbl 0619.60074

Stochastic processes, 6th Semin., Charlottesville/Va. 1986, Prog. Probab. Stat. 13, 139-147 (1987).
[For the entire collection see Zbl 0607.00014.]
The notion of packing measure was introduced by S. J. Taylor and C. Tricot [Trans. Am. Math. Soc. 288, 679-699 (1985; Zbl 0537.28003)], as a new tool for measuring the size of subsets of \({\mathbb{R}}^ d\). The present paper deals with the packing measure of a planar Brownian path (the case of Brownian motion in \({\mathbb{R}}^ d\), \(d\geq 3\), was treated by Taylor and Tricot). An integral test is given which determines, for every measure function f, whether the f-packing measure of the path is zero or infinite. In particular, there exists no measure function f such that the corresponding measure is both positive and finite. The proof depends on a simple representation for the local time process of a two-dimensional Bessel process.

MSC:

60J65 Brownian motion
60J55 Local time and additive functionals
60G17 Sample path properties