Tightness for Thick Points in two dimensions

TL;DR

This paper investigates the tightness of the maximum local time of planar Brownian motion near points in the unit disk, revealing precise asymptotic behavior involving logarithmic corrections.

Contribution

It establishes the tightness of the centered and scaled maximum local time for two-dimensional Brownian motion, providing new insights into its fine asymptotic structure.

Findings

Proves tightness of the scaled maximum local time in two dimensions.

Identifies the precise logarithmic correction terms involved.

Enhances understanding of thick points in planar Brownian motion.

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.