Spectrum of Ornstein-Uhlenbeck operators in \(L ^{p}\) spaces with respect to invariant measures
type: Item
instance of: scholarly article
title: Spectrum of Ornstein-Uhlenbeck operators in \(L ^{p}\) spaces with respect to invariant measures
zbMATH Open document ID: 1027.47036
published in: Journal of Functional Analysis
publication date: 2003-03-26T00:00:00Z
review text: Let \(A:= \sum q_{ij} D_{ij}+ \sum b_{ij} X_j D_i\) denote a generator of an Ornstein-Uhlenbeck process on a finite-dimensional real vector space \(\mathbb{R}^N\), and denote by \((T(t))_{t\geq 0}\) the corresponding Markov semigroup. Put \(B= (b_{ij})\). If \(\text{Spec}(B)\subseteq \{\text{Re }\lambda< 0\}\), i.e. \((e^{-tB})_{t> 0}\) is contractive, then there exists an invariant measure denoted by \(\mu\). \((T(t))\) defines constraction operator semigroups on \(L^p_\mu= L^p(\mathbb{R}^N, \mu)\) for \(1\leq p< \infty\), with infinitesimal generators \((A_p, D_p)\). The aim of the paper under review is to determine the spectrum of \((A_p, D_p)\). For \(1< p<\infty\), the spectrum is discrete (since \(A_p\) has a compact resolvent) and is determined by the spectrum of \(B\), \(\text{Spec}(A_p)= \{\sum n_i\lambda_i: n_i\in \mathbb{N}\}\) where \(\text{Spec}(B)= \{\lambda_i\}\). The eigenfunctions are polynomials with span dense in \(L^p_\mu\) (cf. Theorem 3.1). In fact, \(\text{Spec}(A_p)\) depends only on the drift term \(L\) of \(A\), an observation which allows (in Section 4) to determine the algebraic multiplicity of eigenvalues \(\gamma\) of \(A_p\). Section 5 shows that the situation is completely different for \(p= 1\). In this case (Theorem 5.1), \(\text{Spec}(A_p)\) coincides with the left half-plane of \(\mathbb{C}\).
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 1888364
zbMATH Keywords: Ornstein-Uhlenbeck operators
zbMATH Keywords: invariant measures
zbMATH Keywords: transition semigroups
zbMATH Keywords: Ornstein-Uhlenbeck process
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: spectrum
author: Giorgio Metafune
author: Diego Pallara
author: E. Priola
reviewed by: Wilfried Hazod
MaRDI profile type: Publication
full work available at URL: jfan.2002.3978
OpenAlex ID: W1979411153
cites work: Gaussian estimates and interpolation of the spectrum in \(L^ p\)
cites work: Q4309937
cites work: Generalized Mehler semigroups and applications
cites work: Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces
cites work: Nonsymmetric Ornstein-Uhlenbeck semigroup as second quantized operator
cites work: Generalized Ornstein-Uhlenbeck semigroups: Littlewood-Paley-Stein inequalities and the P. A. Meyer equivalence of norms
cites work: On the Ornstein-Uhlenbeck operator in spaces of continuous functions
cites work: Stochastic Equations in Infinite Dimensions
cites work: Ergodicity for Infinite Dimensional Systems
cites work: Q4041031
cites work: \(L^ 1\)-properties of intrinsic Schrödinger semigroups
cites work: One-Parameter Semigroups for Linear Evolution Equations
cites work: Analyticity of transition semigroups and closability of bilinear forms in Hilbert spaces
cites work: On analyticity of Ornstein-Uhlenbeck semigroups
cites work: Hypoelliptic second order differential equations
cites work: Q4322913
cites work: On the Ornstein-Uhlenbeck operator in 𝐿² spaces with respect to invariant measures
cites work: Q4364774
cites work: Q4799185
cites work: Q5699173
cites work: Q3938964
cites work: Q3836496
cites work: How violent are fast controls?
cites work: Mathematical control theory: an introduction
DOI: 10.1006/JFAN.2002.3978
description: scientific article; zbMATH DE number 1888364