On Brownian disconnection exponents
type: Item
instance of: scholarly article
title: On Brownian disconnection exponents
zbMATH Open document ID: 0845.60083
DOI: 10.2307/3318489
author: Wendelin Werner
published in: Bernoulli
publication date: 1996-09-16T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bj/1193758712
review text: The paper starts with a short, intuitive revision of the derivation of the strictly positive constant \(\gamma\), the disconnection exponent for the two-dimensional Brownian motion. The right-hand side of the well-known double inequality \({1\over 2\pi} \leq \gamma \leq {1\over 2}\) is improved by the estimate \(\gamma \leq {1\over 2} - {1\over 2} \biggl( {\log 2\over \pi}\biggr)^2 < 0.47566\). A comprehensive outline of the ideas and methods involved in the proof of this new upper bound for \(\gamma\) is given separately from the technical details of the proof. Furthermore, the arguments of this proof are adapted to verify the following upper bound for the disconnection exponent \(\gamma_n\) (corresponding to the event that the union of \(n\) independent planar Brownian motion paths, stopped at their respective hitting times of a large circle, does not disconnect zero from infinity): \(\gamma \leq {n \over 2} - {(\log 2)^2\over 2\pi^2 n}\) for all \(n > 1\). A well written derivation of \(\gamma_n\) precedes the proof of this second proposition. Using the similarities between disconnection and intersection exponents, some of the previously used arguments for disconnection exponents are modified to also provide some improved upper bounds for intersection exponents of planar Brownian motion. Several interesting conjectures regarding disconnection and intersection exponents are motivated throughout the paper.
reviewed by: Vera R. Eastwood
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J50
zbMATH DE Number: 846993
zbMATH Keywords: Brownian motion
zbMATH Keywords: disconnected exponent
zbMATH Keywords: intersection exponent
MaRDI profile type: Publication
OpenAlex ID: W4238868235
description: scientific article; zbMATH DE number 846993