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On the most visited sites of planar Brownian motion. (English) Zbl 1244.60079

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)].

MSC:

60J65 Brownian motion
60G17 Sample path properties

Citations:

Zbl 0622.60021
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