Self-intersection local time of order \(k\) for Gaussian processes in \({\mathcal S}^ \prime (\mathbb R^ {d})\).
type: Item
instance of: scholarly article
title: Self-intersection local time of order \(k\) for Gaussian processes in \({\mathcal S}^ \prime (\mathbb R^ {d})\).
zbMATH Open document ID: 1059.60050
DOI: 10.1016/S0304-4149(01)00099-0
author: Anna Talarczyk
published in: Stochastic Processes and their Applications
publication date: 2005-02-25T00:00:00Z
review text: Let \((X_ {t})_ {t\in [0,1]}\) be a centered continuous Gaussian process in \({\mathcal S}^ \prime (\mathbb R^ {d})\) (the space of tempered distributions on \(\mathbb R^ {d}\)). Its self-intersection local time of order \(k\) is, non-rigorously, defined by \(\int _ {[0,t]^ {k}}\langle X_ {s_ 1}\otimes \cdots \otimes X_ {s_ {k}},\varphi (x_ 1)\delta (x_ 2 -x_ 1)\cdots \delta ( x_ {k}-x_ 1)\rangle \, ds_ 1\cdots ds_ {k}\), where \(\delta \) is the Dirac distribution, \(\varphi \in {\mathcal S} (\mathbb R^ {d})\), and \(\otimes \) denotes the tensor product in \({\mathcal S}^ \prime (\mathbb R^ {d})\). More precisely: define \(\:\!X_ {s_ 1}\otimes \cdots \otimes X_ {s_ {k}}\:\) by the relation \[ \langle :X_ {s_ 1}\otimes \cdots \otimes X_ {s_ {k}}:,\;\varphi _ 1\otimes \cdots \otimes \varphi _ {k}\rangle =\! :\langle X_ {s_ 1},\varphi _ 1\rangle \cdots \langle X_ {s_ {k}}, \varphi _ {k}\rangle\!:, \] the right-hand side being the Wick product of the jointly Gaussian variables \(\langle X_ {s_ {i}},\varphi _ {i}\rangle \). Let \({\mathcal F} \subset {\mathcal S}(\mathbb R^ {d})\) be the subset of all symmetric nonnegative functions \(f\) vanishing outside the unit ball of \(\mathbb R ^ {d}\) and satisfying \(\int _ {\mathbb R^ {d}} f\, dx=1\). For \(f\in {\mathcal F}\), \(\varepsilon >0\) and \(\varphi \in {\mathcal S}(\mathbb R^ {d})\) set \[ \Phi _ {f,\varepsilon ,\varphi }(x_ 1,\ldots ,x_ {k})= {1\over \varepsilon ^ {d(k-1)}} \varphi (x_ 1)f\Bigl ({x_ {2}-x_ {1}\over \varepsilon }\Bigr )\cdots f\Bigl ({x_ {k}-x_ {1}\over \varepsilon }\Bigr ), \] and define an \({\mathcal S}^ \prime (\mathbb R^ {d})\)-valued process \(L_ {f,\varepsilon }(t)\) by \[ \langle L_ {f,\varepsilon }(t), \varphi \rangle = \int _ {[0,t]^ {k}} \langle :\!X_ {s_ 1}\otimes \cdots \otimes X_ {s_ {k}}\;1:,\;\Phi _ {f, \varepsilon ,\varphi }\rangle \, ds_ {1}\cdots ds_ {k}, \quad \varphi \in {\mathcal S}(\mathbb R^ {d}). \] An \({\mathcal S}^ \prime (\mathbb R^ {d})\)-valued process \(L(t)\) is ca…
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 2138903
zbMATH Keywords: tempered distributions
zbMATH Keywords: Gaussian processes
zbMATH Keywords: self-intersection local time
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2138903