Smoothness of the convex hull of planar Brownian motion
type: Item
instance of: scholarly article
title: Smoothness of the convex hull of planar Brownian motion
zbMATH Open document ID: 0678.60073
author: S. H. Smith
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: Let B(t) be a two-dimensional Brownian motion and let C(t) be the closed convex hull of the range of B on the time interval [0,t]. The authors prove that, for a given \(t>0\), with probability one, C(t) has no corners and therefore its boundary \(\partial C(t)\) is a \(C^ 1\)-curve. This fact was already stated, with a somehow non-rigorous proof, by \textit{P. Lévy} in his book ``Processus stochastiques et mouvement brownien'' (1965; Zbl 0137.116), and was rigorously proved by \textit{El Bachir} [Thèse, Univ. Toulouse (1983)]. In the present paper, this result is proved anew, and the authors also obtain some (negative) information on the modulus of continuity of the slope of the tangent to \(\partial C(t)\). Related results have been recently obtained by \textit{K. Burdzy} and \textit{L. San Martin} [Stochastic Processes Appl. (to appear)].
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 4111736
zbMATH Keywords: planar Brownian motion
zbMATH Keywords: convex hull
zbMATH Keywords: modulus of continuity
reviewed by: J.-F. Le Gall
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991500
OpenAlex ID: W2035578843
DOI: 10.1214/AOP/1176991500
description: scientific article; zbMATH DE number 4111736