Thick points for the Cauchy process

Abstract

Let $\mathcal{T}(x,\eps)$ denote the occupation measure of an interval of length $2\eps$ centered at $x$ by the Cauchy process run until it hits $(-\infty ,-1]\cup [1,\infty )$. We prove that $\sup_{|x|\leq 1}\mathcal{T}(x,\eps)/(\eps(\ln\eps)^2)\to 2/π$ a.s. as $\eps\to 0$. We also obtain the multifractal spectrum for thick points, i.e. the Hausdorff dimension of the set of $\alpha$-thick points $x$ for which $\lim_{\eps \to 0} \mathcal{T}(x,\eps)/(\eps(\ln\eps)^2) = α> 0$.

16 pages