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Hausdorff dimension of cut points for Brownian motion. (English) Zbl 0891.60078

Summary: Let \(B\) be a Brownian motion in \(\mathbb{R}^d\), \(d=2,3\). A time \(t\in [0,1]\) is called a cut time for \(B[0,1]\) if \(B[0,t) \cap B(t,1]=\emptyset\). We show that the Hausdorff dimension of the set of cut times equals \(1 - \zeta\), where \(\zeta = \zeta_d\) is the intersection exponent. The theorem, combined with known estimates on \(\zeta_3\), shows that the percolation dimension of Brownian motion (the minimal Hausdorff dimension of a subpath of a Brownian path) is strictly greater than one in \(\mathbb{R}^3\).

MSC:

60J65 Brownian motion
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