Measuring close approaches on a Brownian path
Let X be a d-dimensional strictly stable process of index \(\alpha\), \((\alpha =2\) fo a BM), and consider \[ W(h):=\inf_{s,t}\{| X_ t- X_ s|:\quad 0\leq s<t\leq 1,\quad t-s\geq h\} \] for a given \(h>0\). It is proved that in the case \(d>2\alpha\) (for \(d=2\alpha\) see the paper) \[ \liminf_{h\downarrow 0} W(h)/\psi (h)=+\infty \quad a.s.\quad or\quad 0\quad a.s., \] where \(\psi\) is an appropriate testfunction, according to \[ \int_{o+}\psi (h)^{d-2\alpha}h^{-d/\alpha}dh<\infty \quad or\quad =\infty. \] This is a result on the uniform two-sided escape rate of X, and is compared with the corresponding result for a fixed time [see \textit{N. C. Jain}, and \textit{S. J. Taylor}, Ann. Probab. 1, 527-549 (1973; Zbl 0261.60053)]. ``As one would expect, there is a gap between the one- or two-sided local escape rate at a fixed time and the uniform escape rate. We determine the Hausdorff dimension of various sets of unusually slow one- or two-sided escape.''
scientific article; zbMATH DE number 4078445