Dimensional properties of one-dimensional Brownian motion. (English) Zbl 0672.60077

Kaufman, Robert

Ann. Probab. 17, No. 1, 189-193 (1989).

It is shown that for almost all paths in the sample space of the one- dimensional Brownian motion the following statements are true:
1.) For each closed set \(F\subseteq [0,1]\), dim X(F\(+t)=\min (1\), 2 dim F) for almost all \(t>0.\)
2.) For each closed set \(F\subseteq [0,1]\), \(X(F+t)\) has strictly positive Lebesgue measure for almost all \(t>0.\)
The proofs are based on appropriate upper bounds for quantities of the following form (in particular, it seems to be notable that a time integral is involved): \[ E[(\int^{1}_{0}\phi (X(x+t)-X(y+t))dt)^ p] \] for any positive integer p and suitable functions \(\phi\).

Reviewer: P.Kröger

MSC:

60J65	Brownian motion
28A75	Length, area, volume, other geometric measure theory
60G17	Sample path properties

Keywords:

Brownian motion