|
Abstract : |
Consider a planar Brownian motion run for nite time. The frontier or \outer boundary " of the path is the boundary of the unbounded component of the complement. Burdzy (1989) showed that the frontier has innite length. We improve this by showing that the Hausdor dimension of the frontier is strictly greater than 1. (It has been conjectured that the Brownian frontier has dimension 4=3, but this is still open.) The proof uses Jones's Traveling Salesman Theorem and a self-similar tiling of the plane by fractal tiles known as Gosper Islands., |
