Tightness for thick points in two dimensions

Let Wt be Brownian motion in the plane started at the origin and let & theta; be the first exit time of the unit disk D1. Let Z 0 1 & mu;0(x, ⠏) = 1{B(x,E)}(Wt) dt, & pi;⠏2 0 and set & mu;*0(⠏) = SUPx & ISIN;D1 & mu;0(x, ⠏). We show that � � � � & mu;* 0(⠏) - 2/& pi; log ⠏-1 - 2 1 log log ⠏-1 is tight.

Automated summary

This article studies the Brownian motion in the plane starting at the origin and defines the first exit time from the unit disk D1 as θ. By defining different measures, an important result is obtained.