On the most visited sites of planar Brownian motion

DOI10.1214/ECP.V17-1809zbMATH Open1244.60079arXiv1202.1847OpenAlexW2140518831MaRDI QIDQ428743

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Publication date: 22 June 2012

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

Full work available at URL: https://arxiv.org/abs/1202.1847

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