On the most visited sites of planar Brownian motion

TL;DR

This paper investigates the size of the set of points visited by planar Brownian motion using gauge functions, resolving a longstanding open problem by characterizing when these sets have positive or zero measure.

Contribution

It provides a complete characterization of the measure of the set of points visited by planar Brownian motion with respect to specific gauge functions, answering a question posed in 1986.

Findings

For α<1, the set of points visited has positive measure.

For α>1, the set of points visited has zero measure.

The result resolves a longstanding open problem in the field.

Abstract

Let (B_t : t > 0) be a planar Brownian motion and define gauge functions $\phi_\alpha(s)=log(1/s)^{-\alpha}$ for $\alpha>0$. If $\alpha<1$ we show that almost surely there exists a point x in the plane such that $H^{\phi_\alpha}({t > 0 : B_t=x})>0$, but if $\alpha>1$ almost surely $H^{\phi_\alpha} ({t > 0 : B_t=x})=0$ simultaneously for all $x\in R^2$. This resolves a longstanding open problem posed by S.,J. Taylor in 1986.