Thick points for the Cauchy process

Let $\mathcal{T}(x,\text{eps})$ denote the occupation measure of an interval of length $2\text{eps}$ centered at $x$ by the Cauchy process run until it hits $(-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$. We prove that $\sup_{|x|\leq 1}$\mathcal{T}$(x,\eps)/(\eps(\ln\eps)^2)\to 2/\pi$a\mathrm{.}s\mathrm{.}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) = \alpha > 0$.