Multiple points on the Brownian frontier

Más información sobre el libro

Brownian sample paths, especially in the critical planar case, exhibit a fascinatingly rich fractal structure. One particular, well-studied subset of the planar Brownian path is its outer boundary, or frontier. A major breakthrough in the analysis of the Brownian frontier was permitted by the introduction of the Schramm-Löwner Evolution (SLE), a oneparameter family of conformally invariant planar processes introduced by Oded Schramm, and its investigation by Lawler, Schramm, Werner and others. In particular, the analogy between the Brownian frontier and SLE with parameter 6 allowed for the precise determination of the disconnection exponents for planar Brownian motion. The present work focuses on multiply visited points on the Brownian frontier and multiple pioneer points, i. e. points which are multiple frontier points for only a very short time, and determines their Hausdorff dimension in terms of the disconnection exponents. This covers and extends existing results by Lawler, for multiplicity one, and Burdzy and Werner, for frontier points of multiplicity at least three and pioneer points of multiplicity at least four.