The dimension of the Brownian frontier is greater than 1

Details The dimension of the Brownian frontier is greater than 1 The dimension of the Brownian frontier is greater than 1

1995-08-02

arxiv.org/abs/math/9508222v1

Mathematics

Probability

Scientific paper

Consider a planar Brownian motion run for finite 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 infinite length. We improve this by showing that the Hausdorff 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.

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