The Dimension of the Brownian Frontier is Greater Than 1
نویسندگان
چکیده
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 innnite length. We improve this by showing that the Hausdorr 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.
منابع مشابه
2 A ug 1 99 5 The Dimension of the Brownian Frontier is Greater Than 1
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 ...
متن کاملThe Frontier of a Brownian Path Is Multifractal
We consider the multifractal spectrum of harmonic measure of a Brownian motion path in two or three dimensions. We show that the multifractal spectrum is nontrivial and relate the spectrum to the intersection exponent. As a corollary we show that harmonic measure on a three dimension Brownian motion path is carried on a set of Hausdorr dimension strictly less than two.
متن کاملThe Hausdorff dimension of the double points on the Brownian frontier
The frontier of a planar Brownian motion is the boundary of the unbounded component of the complement of its range. In this paper we find the Hausdorff dimension of the set of double points on the frontier. Résumé: Nous déterminons la dimension de Hausdorff de l’ensemble des points doubles situés sur la frontière d’un mouvement brownien plan. MSC 2000: Primary 60J65; Secondary 60G17.
متن کاملThe Frontier of a Brownian
We consider the multifractal spectrum of harmonic measure of a Brownian motion path in two or three dimensions. We show that the multifractal spectrum is nontrivial and relate the spectrum to the intersection exponent. As a corollary we show that harmonic measure on a three dimension Brownian motion path is carried on a set of Hausdorr dimension strictly less than two.
متن کاملBoundary Behavior of Sle
Introduction. Several lattice models from statistical physics such as random walks (RWs), loop-erased random walks (LERWs), self-avoiding random walks (SAWs), and critical FK (Fortuin and Kasteleyn) percolations have been shown or are conjectured to be invariant under conformal mappings. The stochastic Loewner evolution (SLE) was first introduced by O. Schramm as a possible scaling limit for th...
متن کامل