Tightness for thick points in two dimensions

Abstract

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

$${\mathrm{\mu }}_{\mathit{\theta }}(x,\mathit{ϵ})=\frac{1}{\mathrm{\pi }{\mathit{ϵ}}^2}{\int }_0^{\mathit{\theta }}{1}_{\{B(x,\mathit{ϵ})\}}(W_t)dt,$$

and set ${\mathrm{\mu }}_{\mathit{\theta }}^{\ast }(\mathit{ϵ})={sup}_{x\in D_1}{\mathrm{\mu }}_{\mathit{\theta }}(x,\mathit{ϵ})$. We show that

$$\sqrt{{\mathrm{\mu }}_{\mathit{\theta }}^{\ast }(\mathit{ϵ})}-\sqrt{2∕\mathrm{\pi }}\left(log{\mathit{ϵ}}^{-1}-\frac{1}{2}loglog{\mathit{ϵ}}^{-1}\right)$$

is tight.