The dimension of the frontier of planar Brownian motion
Let \(B\) be a two-dimensional Brownian motion, and define the frontier \(F\) of the unbounded connected component of the complement of \(\{B_t; t\in[0,1]\}\). By analogy with conjectures on self-avoiding walks, \textit{B. B. Mandelbrot} [``The fractal geometry of nature'' (1982; Zbl 0504.28001)] conjectured that the Hausdorff dimension of \(F\) is 4/3. The author shows that this Hausdorff dimension can be expressed in terms of the two-sided disconnection exponent \(\alpha\), which is a constant measuring the decay of the probability that the union of two Brownian paths disconnect a fixed point from infinity, when times increases. More precisely, it is shown that the Hausdorff dimension of \(F\) is exactly \(2(1-\alpha)\). Combining this with an estimate of \(\alpha\) derived by the reviewer [Electron. Commun. Probab. 1, 19-28 (1996)], this implies that the dimension of \(F\) is larger than 1.015 (which is larger than 1). The proof relies on highly nontrivial estimates on the asymptotics for non-disconnection probabilities. The approach is similar to that used by the author [Electron. J. Probab. 1, Paper No. 2].
scientific article; zbMATH DE number 918333