Chaos expansions of double intersection local time of Brownian motion in \(\mathbb{R}^ d\) and renormalization
type: Item
instance of: scholarly article
title: Chaos expansions of double intersection local time of Brownian motion in \(\mathbb{R}^ d\) and renormalization
zbMATH Open document ID: 0822.60048
DOI: 10.1016/0304-4149(94)00041-Q
author: P. Imkeller
author: Víctor Pérez-Abreu
author: Josep Vives
published in: Stochastic Processes and their Applications
publication date: 1995-05-23T00:00:00Z
review text: Let \(W = \{W_ t,\;t \in [0,1]\}\) be a \(d\)-dimensional Brownian motion. Denoting by \(p^ d_ \varepsilon\) the probability density of \(W_ \varepsilon\), the authors derive the representation of \[ \alpha_ \varepsilon (x) = \int^ 1_ 0 \int^ 1_ 0 p^ d_ \varepsilon (W_ t - W_ s - x) dsdt, \qquad x \in R^ d,\;\varepsilon > 0, \] by series of multiple Wiener-Itô integrals for any dimension \(d\), and from this they deduce that \(\alpha(x) = \lim_{\varepsilon \downarrow 0} \alpha_ \varepsilon (x)\) can be considered as a well-defined object in the Sobolev space \(D^{2,\gamma}\) over the canonical Wiener space of any order \(\gamma < (4-d)/2\) if \(x \neq 0\). From here they rederive the existence of the double intersection local time for \(d = 2,3\), shown by \textit{J. Rosen} [Ann. Probab. 14, 1245-1251 (1986; Zbl 0617.60079), Stochastic Processes Appl. 23, 131-141 (1986; Zbl 0612.60070), and in: Séminaire de probabilités XX, Lect. Notes Math. 1204, 515-531 (1986; Zbl 0611.60065)]; for \(d\geq 4\) the results are new and could be compared with the smoothed versions considered by \textit{M. Yor} [in: Séminaire de probabilités XIX, Lect. Notes Math. 1123, 332-349 (1985; Zbl 0563.60073)]. For \(x \to 0\), \(\alpha(x)\) does not converge in any Sobolev space, \(\alpha\) has to be renormalised by a suitable deterministic function in order to get the convergence. The difficulty consists in a control of the norms of the projections on the eigenspaces of the Ornstein-Uhlenbeck operator. In case \(d = 2\), \textit{S. R. S.Varadhan's} renormalization (1969) is improved, in case \(d = 3\), versions of \textit{M. Yor's} renormalization (loc. cit.) turn up, and also the case \(d \geq 4\) is dealt with.
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 757547
zbMATH Keywords: renormalization
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: Brownian motion
zbMATH Keywords: multiple Wiener- Itô integrals
zbMATH Keywords: intersection local time
zbMATH Keywords: Ornstein-Uhlenbeck operator
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
cites work: Dirichlet forms and analysis on Wiener space
cites work: Q5610799
cites work: Polynomials of the occupation field and related random fields
cites work: Regularized self-intersection local times of planar Brownian motion
cites work: Self-intersection gauge for random walks and for Brownian motion
cites work: A local time analysis of intersections of Brownian paths in the plane
cites work: Q3209936
cites work: Stochastic Integration for Some Rough Non‐adapted Processes
cites work: The asymptotic behaviour of local times and occupation integrals of the \(N\)-parameter Wiener process in \(\mathbb{R}^ d\)
cites work: Q3672848
cites work: Q3676927
cites work: Sur la saucisse de Wiener et les points multiples du mouvement Brownien. (On the Wiener sausage and the multiple points of Brownian motion)
cites work: Q3749875
cites work: Smoothness of Brownian local times and related functionals
cites work: Chaos expansions and local times
cites work: On the existence of self-intersections for quasi-every Brownian path in space
cites work: A local time approach to the self-intersections of Brownian paths in space
cites work: Tanaka's formula and renormalization for intersections of planar Brownian motion
cites work: Q3750740
cites work: White noise analysis and Tanaka formula for intersections of planar Brownian motion
cites work: The local time of self-intersections of Brownian motions as generalized Brownian functionals
cites work: Q3138660
cites work: Lectures on stochastic differential equations and Malliavin calculus
cites work: On Edwards' model for long polymer chains
cites work: Wiener path intersections and local time
cites work: Q3676928
cites work: Q3685793
full work available at URL: 0304-4149(94)00041-q
OpenAlex ID: W2087893767
description: scientific article; zbMATH DE number 757547