Intersection local times of independent Brownian motions as generalized white noise functionals
type: Item
instance of: scholarly article
title: Intersection local times of independent Brownian motions as generalized white noise functionals
zbMATH Open document ID: 1033.60081
DOI: 10.1023/A:1014212906782
author: L. Streit
author: S. Albeverio
author: M. J. Oliveira
published in: Acta Applicandae Mathematicae
publication date: 2002-05-21T00:00:00Z
review text: \textit{M. De Faria, T. Hida, L. Streit} and \textit{H. Watanabe} [Acta Appl. Math. 46, 351--362 (1997; Zbl 0876.60068)] considered the self-intersection local time of a \(d\)-dimensional Brownian motion. Similarly, the intersection local time of two independent \(d\)-dimensional Brownian motions \(B^{(1)}\) and \(B^{(2)}\) can be defined for \(d<4\) using Donsker's \(\delta\)-function, by \(L\equiv \int d^2t\delta(B^{(1)}(t_1)-B^{(2)}(t_2))\). Further procedures will make \(L\) a well-defined generalized function of Brownian motion for all \(d\geq 1\). However, the chaos expansion of \(L\) is done in terms of `normal product' of white noise and it still corresponds to that in terms of the multiple Wiener integrals when one considers the Wiener process as the fundamental random variable. For all \(d\geq 1\) and \(\varepsilon>0\), the kernel functions of the intersection time \(L_{\varepsilon}\) are given explicitly. For the projection onto chaos of order \(n\geq N\) with \(2N>d-4\), the \(L_p\) properties of the kernel functions \(G_{m,k}\) appearing in the expansion of the truncated local time \(L^{(N)}\) are considered. It is shown in particular that for \(d<4\) they are square-integrable on \(R^{m+k}\) whenever \(m\neq 0\), \(k\neq 0\).
reviewed by: Daniel Boivin
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 46F25
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 82D60
zbMATH DE Number: 1742913
zbMATH Keywords: intersection local times
zbMATH Keywords: chaos expansion
zbMATH Keywords: Brownian motions
zbMATH Keywords: white noise functionals
zbMATH Keywords: polymers
zbMATH Keywords: quantum fields
zbMATH Keywords: Donsker's delta-function
zbMATH Keywords: Hida distributions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1742913