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The dimension of the Brownian frontier is greater than 1. (English) Zbl 0870.60077
The main result states loosely speaking that random (statistically self-similar) compact connected sets \(K\), which possess a certain ‘wigglyness’ condition (uniformly at all scales) have a Hausdorff dimension that is strictly larger than 1. The proof uses several ingeneous ideas such as a random tree associated to a self-similar tiling of the plane (and of \(K\)) by fractal tiles (the so-called Gosper islands) and P. W. Jones’ traveling salesman theorem [Invent. Math 102, No. 1, 1-15 (1990; Zbl 0731.30018)]. This result is the ‘random counterpart’ of a result of the first two authors for deterministic ‘wiggly’ compact connected sets [“Wiggly sets and limit sets”, Preprint]. The randomness yields important additional difficulties in the proof. A striking example of a set satisfying the above-mentioned uniform wigglyness condition is the so-called frontier of planar Brownian motion (more precisely: the boundary of the unbounded connected component of the complement of planar Brownian motion run for finite time). Therefore, the authors derive the result stated in the title. It has been conjectured and it seems to be confirmed by simulations that the exact Hausdorff dimension of the frontier of planar Brownian motion is 4/3, but this is still open. For another approach with recent related results and simulations concerning the frontier of planar Brownian motion, see the papers by G. Lawler [Electron. Commun. Probab. 1, 29-47 (1996; Zbl 0857.60083)] and the reviewer [ibid. 1, 19-28 (1996; Zbl 0862.60069)] and G. Lawler [“The frontier of a Brownian path is multifractal” (Duke Math. preprint, 97-03)].
Reviewer: W.Werner (Paris)

MSC:
60J65 Brownian motion
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