On the most visited sites of planar Brownian motion

Abstract

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