Ornstein-Uhlenbeck semi-groups on stratified groups
type: Item
instance of: scholarly article
title: Ornstein-Uhlenbeck semi-groups on stratified groups
zbMATH Open document ID: 1192.47042
author: Françoise Lust-Piquard
published in: Journal of Functional Analysis
publication date: 2010-03-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.1143
review text: Two analogues of the Ornstein-Uhlenbeck semigroup in the setting of stratified groups \(G\) are constructed, namely, Markovian diffusion semigroups acting on \(L^q(pd\gamma)\), where the invariant density \(p\) is a heat kernel at time 1 on \(G\). The first semigroup is symmetric on \(L^2(pd\gamma)\) with generator \(\sum^n_{i=1} X_i* X_i,\) where \((X_i)^n_{i=1}\) is a linear basis of the first layer of the Lie algebra of \(G\) and \(X_i*\) the adjoint of \(X_i\) with respect to \(L^2(pd\gamma)\), \(i= 1,\dots,n\). The second semigroup, defined by an analogue of Mehler's formula, is compact on \(L^q(pd\gamma)\), \(1< q<\infty\) and non-symmetric on \(L^2(pd\gamma)\). The formal real part of its generator \(N\) is \(\sum^n_{i=1} X_i* X_i.\) Both generators have the same caré du champ, the difference between them being an anti-symmetric first-order differential operator. If polynomials are dense in \(L^2(pd\gamma)\), i.e., if \(G\) has at most four layers, the spectrum of \(N\) is seen to be the set of nonnegative integers, and the corresponding set of eigenspaces is determined. For the case that \(G\) is step two, another description of the eigenspaces is given, resembling the classical definition of Hermite polynomials via their generating function.
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60B99
zbMATH DE Number: 5676155
zbMATH Keywords: stratified groups
zbMATH Keywords: sub-Laplacian
zbMATH Keywords: heat kernel measure
zbMATH Keywords: Ornstein-Uhlenbeck semigroups
reviewed by: Heinrich Hering
MaRDI profile type: Publication
OpenAlex ID: W2126667074
arXiv ID: 0907.1143
cites work: On logarithmic Sobolev inequalities. With a preface of Dominique Bakry and Michel Ledoux
cites work: On gradient bounds for the heat kernel on the Heisenberg group
cites work: Ornstein-Uhlenbeck processes on Lie groups
cites work: Q4002250
cites work: Q4164574
cites work: Heat kernels for class 2 nilpotent groups
cites work: Q4041031
cites work: Hypoelliptic heat kernel inequalities on the Heisenberg group
cites work: Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28
cites work: Q4423258
cites work: Coercive inequalities on metric measure spaces
cites work: The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group
cites work: Q4531961
cites work: Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg
cites work: Riesz transforms on deformed Fock spaces
cites work: A simple-minded computation of heat kernels on Heisenberg groups
cites work: Q3744958
cites work: Q5645344
cites work: Harmonic analysis on the Heisenberg group
cites work: Q4527707
DOI: 10.1016/J.JFA.2009.11.012
description: scientific article