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 \[μ_{ θ} ( x,ε) =\frac{1}{πε^{ 2} }\int_{0}^{ θ}1_{\{ B( x,ε)\}}( W_{t})\,dt,\] and set ${\mu }_{\theta }^{\ast }(\epsilon )={\sup }_{x\in D_1}{\mu }_{\theta }(x,\epsilon )$. We show that \[\sqrt{μ^{\ast}_θ (ε)}-\sqrt{2/π} \left(\log ε^{-1}- \frac{1}{2}\log\log ε^{-1}\right)\] is tight.