Let $B$ be a Brownian motion in $R^d$, $d=2,3$. A time $tin [0,1]$ 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 $R^3$.
展开▼
机译：假设$ B $是$ R ^ d $中的布朗运动，$ d = 2,3 $。如果$ B [0，t） cap B（t，1] = emptyset $，则将时间$ t in [0,1] $称为$ B [0,1] $的切割时间。切割时间集的Hausdorff维数等于$ 1- zeta $，其中$ zeta = zeta_d $是交点指数，该定理与$ zeta_3 $的已知估计值相结合，表明布朗运动的渗透维数（布朗路径的子路径的最小Hausdorff维数）严格大于$ R ^ 3 $中的一个。
展开▼
