Hausdorff Dimension of Cut Points for Brownian Motion

Let B be a Brownian motion in R, d = 2, 3. A time t ∈ [0, 1] is called a cut time for B[0, 1] if B[0, t)∩B(t, 1] = ∅. We show that the Hausdorff dimension of the set of cut times equals 1−ζ, where ζ = ζd is the intersection exponent. The theorem, combined with known estimates on ζ3, 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.