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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
##### Keywords:
multifractals analysis
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