The structure of a Brownian bubble
type: Item
instance of: scholarly article
title: The structure of a Brownian bubble
zbMATH Open document ID: 0794.60047
DOI: 10.1007/BF01200206
author: Robert C. Dalang
author: John B. Walsh
publication date: 1994-05-18T00:00:00Z
review text: We examine local geometric properties of level sets of the Brownian sheet, and in particular, we identify the asymptotic distribution of the area of sets which correspond to excursions of the sheet high above a given level in the neighborhood of a particular random point. It is equal to the area of certain individual connected components of the random set \(\{(s,t):B(t)>b(s)\}\), where \(B\) is a standard Brownian motion and \(b\) is (essentially) a Bessel process of dimension 3. This limit distribution is studied and, in particular, explicit formulas are given for the probability that a point belongs to a specific connected component, and for the expected area of a component given the height of the excursion of \(B(t)-b(s)\) in this component. These formulas are evaluated numerically and compared with the results from direct simulations of \(B\) and \(b\).
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 568993
zbMATH Keywords: Brownian sheet
zbMATH Keywords: asymptotic distribution
zbMATH Keywords: standard Brownian motion
zbMATH Keywords: simulations
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01200206
OpenAlex ID: W1976747935
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 568993