The Frontier of a Brownian

The Dimension of the Brownian Frontier is Greater Than 1

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...

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 ...

The 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.

Brownian motion, complex analysis, and the dimension of the Brownian frontier

A simple path to asymptotics for the frontier of a branching Brownian motion

The Dimension of the Planar Brownian Frontier is 4 / 3

The purpose of this note is to announce and sketch the proofs of results determining the Hausdorff dimension of certain subsets of planar Brownian paths. Proofs are currently written down in a sequence of preprints [9, 10, 11, 12]. The present announcement will give an overview of some of the contents of this series. At this point, it appears that some parts of the proofs can be significantly s...