The Annals of Probability

Measuring Close Approaches on a Brownian Path

Edwin A. Perkins and S. James Taylor

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60G17: Sample path properties 60J30

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


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.

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