## The Annals of Probability

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

### Measuring Close Approaches on a Brownian Path

Edwin A. Perkins and S. James Taylor

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60J65: Brownian motion [See also 58J65]

Secondary: 60G17: Sample path properties 60J30

**Keywords**

Brownian motion two-sided escape rate uniform escape rate integral test Hausdorff dimension stable process

#### Citation

Perkins, Edwin A.; Taylor, S. James. Measuring Close Approaches on a Brownian Path. Ann. Probab. 16 (1988), no. 4, 1458--1480. doi:10.1214/aop/1176991578. https://projecteuclid.org/euclid.aop/1176991578