## The Annals of Probability

### Measuring Close Approaches on a Brownian Path

#### Abstract

Integral tests are found for the uniform escape rate of a $d$-dimensional Brownian path $(d \geq 4)$, i.e., for the lower growth rate of $\inf\{|X(t) - X(s)|: 0 \leq s, t \leq 1, |t - s| \geq h\}$ as $h \downarrow 0$. The gap between this uniform escape rate and the one-sided local escape rate of Dvoretsky and Erdos and the two-sided local escape rate of Jain and Taylor suggest the study of certain sets of times of slow one- or two-sided escape. The Hausdorff dimension of these exceptional sets is computed. The results are proved for a broad class of strictly stable processes.

#### Article information

Source
Ann. Probab., Volume 16, Number 4 (1988), 1458-1480.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991578

Digital Object Identifier
doi:10.1214/aop/1176991578

Mathematical Reviews number (MathSciNet)
MR958197

Zentralblatt MATH identifier
0659.60113

JSTOR