## 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

### Keywords:

Hausdorff dimension; percolation dimension; Brownian motion
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