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{1176991578,
author = {Perkins, Edwin A. and Taylor, S. James},
title = {Measuring Close Approaches on a Brownian Path},
journal = {Ann. Probab.},
volume = {16},
number = {4},
year = {1988},
pages = { 1458-1480},
language = {en},
url = {http://dml.mathdoc.fr/item/1176991578}
}
Perkins, Edwin A.; Taylor, S. James. Measuring Close Approaches on a Brownian Path. Ann. Probab., Tome 16 (1988) no. 4, pp. 1458-1480. http://gdmltest.u-ga.fr/item/1176991578/