Geography of the level sets of the Brownian sheet
type: Item
instance of: scholarly article
title: Geography of the level sets of the Brownian sheet
zbMATH Open document ID: 0792.60038
DOI: 10.1007/BF01192131
author: Robert C. Dalang
author: John B. Walsh
publication date: 1994-07-14T00:00:00Z
review text: We describe geometric properties of \(\{W>\alpha\}\), where \(W\) is a standard real-valued Brownian sheet, in the neighborhood of the first hit \(P\) of the level set \(\{W>\alpha\}\) along a straight line or smooth monotone curve \(L\). In such a neighborhood we use a decomposition of the form \(W(s,t)=\alpha-b(s)+B(t)+x(s,t)\), where \(b(s)\) and \(B(t)\) are particular diffusion processes and \(x(s,t)\) is comparatively small, to show that \(P\) is not on the boundary of any connected component of \(\{W>\alpha\}\). Rather, components of this set form clusters near \(P\). An integral test for thorn-shaped neighborhoods of \(L\) with tip at \(P\) that do not meet \(\{W>\alpha\}\) is given. We then analyse the position and size of clusters and individual connected components of \(\{W>\alpha\}\) near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions of \(B\) and \(b\) and an accounting of the error term \(x\).
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 569086
zbMATH Keywords: Brownian sheet
zbMATH Keywords: level set
zbMATH Keywords: clusters
zbMATH Keywords: length of excursions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01192131
OpenAlex ID: W2079960227
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569086