On the most visited sites of planar Brownian motion

Abstract

Let $(B_t \colon t \ge 0)$ 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 ${\mathcal H}^{\phi_\alpha}(\{t \ge 0 \colon B_t=x\})>0$,but if $\alpha>1$ almost surely ${\mathcal H}^{\phi_\alpha} (\{t \ge 0 \colon B_t=x\})=0$ simultaneously for all $x\in{\mathbb R}^2$. This resolves a longstanding open problem posed by S.J. Taylor in 1986.