Abstract
Let $B$ be a Brownian motion in $R^d$, $d=2,3$. 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 $R^3$.
Highlights
Let Bt = B(t) denote a Brownian moton taking values in Rd, (d = 2, 3)
A time t ∈ [0, 1] is called a cut time for B[0, 1] if
Burdzy [1] has shown that nontrivial cut points exists, i.e., with probability one L∩(0, 1) = ∅
Summary
Theorem 1.1 If B is a Brownian motion in Rd, d = 2, 3 and L is the set of cut times of B[0, 1], with probability one, dimh(L) = 1 − ζ, where ζ = ζd is the intersection exponent. Denote the set of cut points on B[0, 1] It is well known [11, 17] that Brownian motion doubles the Hausdorff dimension of sets in [0, 1]. By a standard argument, we get that Jn ≥ c(2n)1−ζ with some positive probability, independent of n This gives a good indication that the Hausdorff dimension of L should be 1 − ζ, and with this bound standard techniques can be applied to establish the result. I would like to thank Ed Perkins for some useful remarks about Hausdorff dimension
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