Intersection local times of all orders for Brownian and stable density processes -- construction, renormalisation and limit laws
type: Item
instance of: scholarly article
title: Intersection local times of all orders for Brownian and stable density processes -- construction, renormalisation and limit laws
zbMATH Open document ID: 0778.60054
author: Robert J. Adler
author: Jay Rosen
published in: The Annals of Probability
publication date: 1993-10-11T00:00:00Z
review text: Let \(X^ 1\), \(X^ 2,\ldots\) be a sequence of independent symmetric stable processes with index \(\alpha\in(0,2]\) started at 0 (the case \(\alpha=2\) corresponds to Brownian motion). Consider a Poisson point process \(\Pi^ \lambda=\{x^ 1,x^ 2,\ldots\}\) with intensity \(\lambda dx\;(\lambda>0)\) and \(\sigma^ 1,\sigma^ 2\ldots\) a sequence of independent Rademacher variables. Introduce the process \(\eta^ \lambda\) valued in the Schwartz space \({\mathcal S}'\) of temperate distributions given by \[ \eta^ \lambda_ t(\varphi)=\lambda^{-1/2}\sum^ \infty_{i=1}\sigma^ i\varphi(x^ i+X^ i_ t)\qquad(\varphi\in{\mathcal S},\;t\geq 0). \] The density process \(\eta\) is the weak limit of \(\eta^ \lambda\) as \(\lambda\to\infty\), and satisfies a stochastic partial differential equation of the type studied by \textit{J. B. Walsh} [École d'été de probabilités de Saint-Flour XIV-1984, Lect. Notes Math. 1180, 265-437 (1986; Zbl 0608.60060)]. The paper under review is concerned with the \(k\)th order intersection local times (ILT) \(\gamma_ k\) of \(\eta\) \((k=2,3,\ldots)\). For simplicity, we will only discuss the case \(k=2\), but the paper has also results for general \(k\). Formally, \(\gamma_ 2\) is defined by \[ \gamma_ 2(t,\varphi)=\int^ t_ 0du\int^ t_ 0dv\int_{\mathbb{R}^ d}\int_{\mathbb{R}^ d} (\eta_ u\times\eta_ v)\delta(x-y)\varphi(x)dxdy, \tag{*} \] where \(\delta\) stands for the Dirac delta function and \(\varphi\) is a function in the Schwartz space \({\mathcal S}\). In order to make a rigorous definition, one introduces first the approximative ILT, \(\gamma_{2,\varepsilon}\), by replacing \(\delta\) in the right-hand side of \((*)\) by \(f_ \varepsilon\), where \(f_ \varepsilon(x)=\varepsilon^{-d}f(x/\varepsilon)\) and \(f\) is a smooth symmetric function with compact support and \(\int f(x)dx=1\). The main concern of the paper is to study the convergence of \(\gamma_{2,\varepsilon}\) as \(\varepsilon\downarrow 0\). When \(d<2\alpha\), \(\gamma_{2,\varepsilon}(…
reviewed by: Jean Bertoin
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60G20
zbMATH DE Number: 428673
zbMATH Keywords: stable processes
zbMATH Keywords: Poisson point process
zbMATH Keywords: intersection local times
zbMATH Keywords: Gaussian white noise
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989283
OpenAlex ID: W2089244261
DOI: 10.1214/AOP/1176989283
description: scientific article; zbMATH DE number 428673