Brownian processes in infinite dimension
type: Item
instance of: scholarly article
title: Brownian processes in infinite dimension
zbMATH Open document ID: 0822.60070
DOI: 10.1007/BF01275589
author: Denis Feyel
author: Arnaud De La Pradelle
published in: Potential Analysis
publication date: 1995-05-02T00:00:00Z
review text: Consider a zero-mean Gaussian measure \(\xi\) on some locally convex space \(E\), and let \(H\) be the Cameron-Martin space of \((E,\xi)\). The authors construct first a Brownian motion \(W\) with values in \(E\) and covariance \(\xi\), and then the stochastic integral \(\int^ t_ 0 \langle \Phi_ s, dW\rangle\) for a generic predictable process \(\Phi\) with values in \(H\) and such that \(\int^ t_ 0 \mathbb{E}(| \Phi_ s |^ 2)ds < \infty\). They prove that every centered square integrable variable which is measurable with respect to \(W\), can be expressed as a stochastic integral, and also obtain a version of Clark's formula. The main purpose of the paper is to solve a stochastic Cauchy problem \(dX_ t = \sqrt {2a} dW_ t - AX_ t dt\), where \(A\) is the infinitesimal generator of an analytic contracting semigroup \((C_ t)\) of operators of \(H\), and \(a\) a closed Hermitian positive operator such that \(a\) and \(A\) commute. Under some technical conditions, the solution of the preceding SDE is \(X_ t (x) = C_ t x + \sqrt{2a} \int^ t_ 0 C_{t-s}dW_ s\).
reviewed by: Jean Bertoin
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 739491
zbMATH Keywords: Gaussian measure
zbMATH Keywords: Cameron-Martin space
zbMATH Keywords: Clark's formula
zbMATH Keywords: stochastic Cauchy problem
zbMATH Keywords: analytic contracting semigroup
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01275589
OpenAlex ID: W1971231892
description: scientific article; zbMATH DE number 739491