Seminar on Stochastic Processes, 1986 pp 139-147 | Cite as

# The Packing Measure of Planar Brownian Motion

Chapter

## Abstract

In a recent paper [4] a new fractal measure φ-p(E) with respect to the monotone function φ(s) was defined,1and it was shown that φ
for every Borel A ⊂ ℝ

_{1}(s) = s^{2}[log log 1/s]^{−1}is the right function for measuring B^{d}(t), the Brownian motion in ℝ^{d}, d ≥ 3, in the sense that there are finite positive constants c_{d}such that$${{\varphi }_{1}}-p\ {{B}^{d}}(A)={{c}_{d}}\left| A \right|a.s.$$

(1)

^{+}= [0,∞), where |A| denotes the Lebesgue measure of A. This new measure φ-p(E), which we called P-packing measure, involves maximising the φ content of a packing by disjoint balls centered in E, with radii at most δ, and taking the limit as δ ↓ 0. In the present note we consider the problem of determining a function φ_{2}which gives the analogue of (1) for the critical case of planar Brownian motion B^{2}(t), which we will henceforth denote by Z(t). In fact, we show that no such φ_{2}exists, and give a test which determines whether φ-p Z[0,1] is 0 or + ∞. This provides a complete solution to Problem 1 of [5].## Keywords

Brownian Motion Sample Path Packing Measure Occupation Measure Brownian Path
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

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## Copyright information

© Birkhäuser Boston 1987