Tightness for Thick Points in two dimensions

Let $W_{t}$ be Brownian motion in the plane started at the origin and let $θ$ 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 $μ^{ \ast}_{ θ} (ε)=\sup_{x\in D_{1}}μ_{ θ} ( x,ε)$. 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.

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