Regularity of invariant measures for a class of perturbed Ornstein-Uhlenbeck operators
type: Item
instance of: scholarly article
title: Regularity of invariant measures for a class of perturbed Ornstein-Uhlenbeck operators
zbMATH Open document ID: 0852.60071
DOI: 10.1007/BF01195918
author: Guiseppe Da Prato
author: Vladimir I. Bogachev
author: Michael Röckner
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1996-12-02T00:00:00Z
review text: Let \(H\) be a real separable Hilbert space with inner product \(\langle \cdot, \cdot \rangle\) and norm \(|\cdot |\), and \(A\) be a nonnegative definite self-adjoint operator on \(H\) such that \(A^{-1}\) is trace-class. Suppose that \(F\) is bounded and Lipschitz continuous. Sobolev space \(W^{1,2} (H,\mu_0)\) is defined as follows: Let \({\mathcal F}C^\infty_b (H)\) be the set of all functions \(\varphi\) on \(H\) of the form \[ \varphi(x) = f(\langle e_1, x\rangle, \dots, \langle e_n,x\rangle), \qquad x \in H, \] where \(n \in \mathbb{N}\), \(f \in C^\infty_b (\mathbb{R}^n)\), and \(\{e_j\mid j \in \mathbb{N}\}\) is an eigenbasis of \(A\) which we fix from now on. For \(\varphi \in {\mathcal F} C^\infty_b (H)\) let \(\varphi'\) denote its Fréchet derivative, and for \(x \in H\) let \(D_x \varphi(x)\) denote the element corresponding to \(\varphi'(x) \in H^*\). Then \(W^{1,2} (H, \mu_0)\) is defined as the completion of \({\mathcal F} C^\infty_b(H)\) with respect to the norm given by \[ |\varphi|^2_{W^{1,2} (H, \mu_0)} = |\varphi|^2_{L^2(H, \mu_0)} + \int_H |D_x \varphi(x)|^2 \mu_0(dx), \] where \(\mu_0\) is Gaussian measure and \(\mu_0 = {\mathcal N}(0, {1\over 2} A^{-1})\). For \(\varphi \in {\mathcal F} C^\infty_b(H)\) let \(\varphi''\) denote its second Fréchet derivative, and for \(x \in H\) let \(D_{xx} \varphi (x)\) denote the corresponding element in \({\mathcal L}(H)\). Note that \(D_{xx} \varphi(x)\) is trace-class and that \[ D_x \varphi = \sum^n_{j=1} \partial_j f(\langle e_1, \cdot\rangle, \dots, \langle e_n, \cdot \rangle) e_j, \] if \(\varphi = f(\langle e_1, \cdot \rangle, \dots, \langle e_n, \cdot\rangle)\) and \(\partial_jf(y) = {\partial f\over \partial y_j}(y)\), \(y \in \mathbb{R}^n\). Hence \(D_x \varphi(x) \in D(A)\) for all \(x \in H\) and we can define \[ {\mathcal A}_0 \varphi(x) = \text{Tr} [D_{xx} \varphi(x)] + \langle x, AD_x \varphi(x)\rangle, \qquad x \in H, \] and \[ {\mathcal A}_F \varphi = {\mathcal A}_0 \varphi(x) + \langle F, …
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 35R15
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 885329
zbMATH Keywords: Lipschitz continuous
zbMATH Keywords: Sobolev space
zbMATH Keywords: Fréchet derivative
zbMATH Keywords: generalized invariant measure
reviewed by: Stanisław Wedrychowicz
MaRDI profile type: Publication
cites work: Classical Dirichlet forms on topological vector spaces --- closability and a Cameron-Martin formula
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full work available at URL: https://doi.org/10.1007/bf01195918
OpenAlex ID: W2004303356
description: scientific article; zbMATH DE number 885329