Selfadjointness of some infinite-dimensional elliptic operators and application to stochastic quantization
type: Item
instance of: scholarly article
title: Selfadjointness of some infinite-dimensional elliptic operators and application to stochastic quantization
zbMATH Open document ID: 0971.47019
author: Guiseppe Da Prato
author: Luciano Tubaro
publication date: 2001-10-18T00:00:00Z
review text: Let \(H\) be a separable Hilbert space. The authors consider the following linear operator \[ K\varphi =L\varphi-\langle C^{1/2}DU(x),C^{1/2}D\varphi\rangle; \qquad {\mathcal D}(K) = {\mathcal D}(L) \cap C^2_b(H), \] where \(D\varphi\) is Fréchet derivative of \(\varphi\), \(L\varphi = {1\over 2} \operatorname {Tr}[CD^2\varphi]+ \langle Ax,D\varphi\rangle\) is the Ornstein-Uhlenbeck operator, \(A<0\) is linear operators on \(H\), \(C=(-A)^{-\varepsilon}\), \(U\) is a mapping from \(H\) to \({\mathbb R}\) such that \(e^{-U}\in L^p(H,\mu), p\geq 1\). If \(U\in W^{1,4}(H,\mu)\), and \((-A)^{-1-\varepsilon}\) is of trace class, it is proved that operator \(K\) is essentally selfadjoint in the space \(L^2(H,\nu)\), where \(\mu\) is the invariant measure associated with \(L\), \(\nu\) is the probability measure \(\nu(dx)=\operatorname {const}e^{-2U(x)} dx\). An application to stochastic quantization is given.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 81S20
zbMATH DE Number: 1547918
zbMATH Keywords: selfadjointness
zbMATH Keywords: infinite-dimensional elliptic operators
zbMATH Keywords: Fréchet derivative
zbMATH Keywords: essentially selfadjoint
zbMATH Keywords: invariant measure
zbMATH Keywords: stochastic quantization
MaRDI profile type: Publication
full work available at URL: s004400000073
OpenAlex ID: W4232868724
DOI: 10.1007/S004400000073
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1547918