Tightness for thick points in two dimensions

Abstract

Let $W_{t}$ be Brownian motion in the plane started at the origin and let $ \theta$ be the first exit time of the unit disk $D_{1}$. Let \[\mu_{ \theta } ( x,\epsilon) =\frac{1}{\pi\epsilon^{ 2} }\int_{0}^{ \theta }1_{\{ B( x,\epsilon)\}}( W_{t})\,dt,\] and set $\mu^{ \ast}_{ \theta } (\epsilon)=\sup_{x\in D_{1}}\mu_{ \theta } ( x,\epsilon)$. We show that \[\sqrt{\mu^{\ast}_{\theta} (\epsilon)}-\sqrt{2/\pi} \left(\log \epsilon^{-1}- \frac{1}{2}\log\log \epsilon^{-1}\right)\] is tight.