Symmetric Ornstein-Uhlenbeck semigroups and their generators
type: Item
instance of: scholarly article
title: Symmetric Ornstein-Uhlenbeck semigroups and their generators
zbMATH Open document ID: 1028.60057
author: Anna Chojnowska-Michalik
author: B. Goldys
publication date: 2003-05-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0205315
review text: The authors consider a linear stochastic differential equation \(dZ(t)= AZ(t)dt+ \sqrt{Q}dW(t)\) on a Hilbert space \(H\) where \(A\) generates a \(C_0\)-semigroup \(\{S(t)\}_{t\geq 0}\), the operator \(Q\) is bounded selfadjoint nonnegative and \(W\) denotes a standard cylindrical Wiener process. Assuming the existence of an invariant measure \(\mu\) for the equation, necessary and sufficient conditions are provided for the transition semigroup \(\{R(t)\}_{t\geq 0}\) to be symmetric in \(L^2(H,\mu)\). This result enables the authors to characterize the domain of the generator of \(\{R(t)\}_{t\geq 0}\) in \(L^p(H,\mu)\) in terms of appropriate Sobolev spaces and to derive a formula for the size of the spectral gap of the generator. Moreover, necessary and sufficient conditions for the transition semigroup to be compact, Hilbert-Schmidt or strong Feller are given in terms of the semigroup \(\{Q^{-1/2}S(t)Q^{1/2}\}_{t\geq 0}\). The results are applied to study an Ornstein-Uhlenbeck process in a chaotic environment.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 1903697
zbMATH Keywords: Ornstein-Uhlenbeck operator
zbMATH Keywords: transition semigroup
zbMATH Keywords: reversible process
zbMATH Keywords: symmetrizing measure
reviewed by: M. Riedle
MaRDI profile type: Publication
OpenAlex ID: W1512127941
arXiv ID: math/0205315
DOI: 10.1007/S004400200222
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article