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Thick points for the Cauchy process. (English) Zbl 1074.60084
Summary: Let \(\mu(x,\varepsilon)\) denote the occupation measure of an interval of length \(2\varepsilon\) centered at \(x\) by the Cauchy process run until it hits \((-\infty,-1]\cup[1,\infty)\). We prove that \(\sup_{| x|\leq 1}\mu(x, \varepsilon)/(\varepsilon (\log \varepsilon)^2)\) \(\to 2/\pi\) a.s. as \(\varepsilon\to 0\). We also obtain the multifractal spectrum for thick points, i.e. the Hausdorff dimension of the set of \(\alpha\)-thick points \(x\) for which \(\lim_{\varepsilon\to 0}\mu (x,\varepsilon)/(\varepsilon(\log \varepsilon)^2)= \alpha>0\).

MSC:
60J55 Local time and additive functionals
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