Cammarota, Valentina; Mörters, Peter On the most visited sites of planar Brownian motion. (English) Zbl 1244.60079 Electron. Commun. Probab. 17, Paper No. 15, 9 p. (2012). Summary: Let \((B_t, t \geq 0)\) 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 \({\mathcal H}^{\phi_\alpha}(\{t \geq 0 \colon B_t=x\})>0\), but if \(\alpha>1\) almost surely \({\mathcal H}^{\phi_\alpha} (\{t \geq 0 \colon B_t=x\})=0\) simultaneously for all \(x\in{\mathbb R}^2\). This resolves a long standing open problem posed by S. J. Taylor [Math. Proc. Camb. Philos. Soc. 100, 383–406 (1986; Zbl 0622.60021)]. Cited in 2 Documents MSC: 60J65 Brownian motion 60G17 Sample path properties Keywords:Brownian motion; Hausdorff dimension; Hausdorff gauge; exact Hausdorff measure; local time; point of infinite multiplicity; random fractal; uniform dimension estimates Citations:Zbl 0622.60021 × Cite Format Result Cite Review PDF Full Text: DOI arXiv