Local times of fractional Brownian sheets
type: Item
instance of: scholarly article
title: Local times of fractional Brownian sheets
zbMATH Open document ID: 1009.60024
author: Y. Xiao
author: Tu-Sheng Zhang
publication date: 2002-12-01T00:00:00Z
review text: Consider a real-valued fractional Brownian sheet \(B^{H}_{0}=\{ B^{H}_{0}(t)\), \(t\in \mathbb{R}^{N}_{+} \}\). Let \(B^{H}\) be the \((N,d)\) Gaussian random field defined by \(B^{H}(t)=(B^{H}_{1}(t), \ldots , B^{H}_{d}(t))\) \((t\in \mathbb{R}^{N}_{+})\), where \(B^{H}_{1}, \ldots , B^{H}_{d}\) are independent copies of \(B^{H}_{0}.\) The authors show the existence and joint continuity of the local times of the Gaussian random field \(B^{H}.\) Many authors have studied the sample path properties of the Brownian sheet and fractional Brownian motion. And most of the studies are established by the facts of the independent increment property of the Brownian sheet and the local nondeterminism of fractional Brownian motion. But, in general, the fractional Brownian sheet \({B}^{H}\) has neither the property of independent increments nor local nondeterminism. Some researchers have investigated the properties of the fractional Brownian motion without independent increment and local nondeterminism. The authors claim that the existence theorem is sharp and its proof is quite different from the proofs for the Brownian sheet and fractional Brownian motion. However, for the joint continuity, the authors only establish a sufficient condition.
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 1837375
zbMATH Keywords: local times
zbMATH Keywords: fractional Brownian sheet
zbMATH Keywords: Gaussian random field
zbMATH Keywords: joint continuity
reviewed by: Tae Il Jeon
MaRDI profile type: Publication
full work available at URL: s004400200210
OpenAlex ID: W2029257959
DOI: 10.1007/S004400200210
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1837375