Ornstein-Uhlenbeck Semigroup Research Articles

Maximal and quadratic Gaussian Hardy spaces

Building on the author’s recent work with Jan Maas and Jan van Neerven, this paper establishes the equivalence of two norms (one using a maximal function, the other a square function) used to define a Hardy space on \mathbb{R}^{n} with the Gaussian measure, that is adapted to the Ornstein–Uhlenbeck semigroup. In contrast to the atomic Gaussian Hardy space introduced earlier by Mauceri and Meda, the h^{1}(\mathbb{R}^{n};d\gamma) space studied here is such that the Riesz transforms are bounded from h^{1}(\mathbb{R}^{n};d\gamma) to L^{1}(\mathbb{R}^{n};d\gamma) . This gives a Gaussian analogue of the seminal work of Fefferman and Stein in the case of the Lebesgue measure and the usual Laplacian.

Comparison inequalities on Wiener space

We define a covariance-type operator on Wiener space: for F and G two random variables in the Gross–Sobolev space D1,2 of random variables with a square-integrable Malliavin derivative, we let ΓF,G≔〈DF,−DL−1G〉, where D is the Malliavin derivative operator and L−1 is the pseudo-inverse of the generator of the Ornstein–Uhlenbeck semigroup. We use Γ to extend the notion of covariance and canonical metric for vectors and random fields on Wiener space, and prove corresponding non-Gaussian comparison inequalities on Wiener space, which extend the Sudakov–Fernique result on comparison of expected suprema of Gaussian fields, and the Slepian inequality for functionals of Gaussian vectors. These results are proved using a so-called smart-path method on Wiener space, and are illustrated via various examples. We also illustrate the use of the same method by proving a Sherrington–Kirkpatrick universality result for spin systems in correlated and non-stationary non-Gaussian random media.

On extension and refinement of the Poincaré inequality

The aim of this paper is to analyze the heat semigroup \({(\mathcal{N}_{t})_{t >0 } = \{e^{t \Delta}\}_{t >0 }}\) generated by the usual Laplacian operator Δ on \({\mathbb{R}^{d}}\) equipped with the d-dimensional Lebesgue measure. We obtain and study, via a method involving some semigroup techniques, a large family of functional inequalities that does not exist in the literature and with the local Poincare and reverse local Poincare inequalities as particular cases. As a consequence, we establish in parallel a new functional and integral inequality related to the Ornstein–Uhlenbeck semigroup.

L 2-theory for non-symmetric Ornstein–Uhlenbeck semigroups on domains

We prove that the mild solution of the stochastic evolution equation \({{d}X(t) = AX(t)\,{d}t + {d}W(t)}\) on a Banach space E has a continuous modification if the associated Ornstein–Uhlenbeck semigroup is analytic on L2 with respect to the invariant measure. This result is used to extend recent work of Da Prato and Lunardi for Ornstein–Uhlenbeck semigroups on domains \({\mathcal{O} \subseteq E}\) to the non-symmetric case. Denoting the generator of the Ornstein–Uhlenbeck semigroup by \({L_\mathcal{O}}\), we obtain sufficient conditions in order that the domain of \({\sqrt{-L_\mathcal{O}}}\) be a first-order Sobolev space.

L 1 ‐SMOOTHING FOR THE ORNSTEIN–UHLENBECK SEMIGROUP

Given a probability density, we estimate the rate of decay of the measure of the level sets of its evolutes by the Ornstein–Uhlenbeck semigroup. The rate is faster than what follows from the preservation of mass and Markov's inequality.

Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup

We study the Ornstein-Uhlenbeck semigroup (Pt)t≥0={exp(tL)}t≥0 generated by the operator Lf(x)=Δf(x)−x⋅∇f(x), on ℝn equipped with the n-dimensional standard Gaussian measure \(\gamma_{n}(\mathrm{d}x)=(\sqrt{2\pi})^{-n}\exp(-|x|^{2}/2)\mathrm{d}x\). By means of a simple method involving essentially a commutation property between the semigroup and the gradient, we describe a large family of optimal integral inequalities with the logarithmic Sobolev and reverse logarithmic Sobolev inequalities as particular cases.

Endpoint estimates for first-order Riesz transforms associated to the Ornstein–Uhlenbeck operator

In the setting of Euclidean space with the Gaussian measure \gamma , we consider all first-order Riesz transforms associated to the infinitesimal generator of the Ornstein–Uhlenbeck semigroup. These operators are known to be bounded on L^p(\gamma) , for 1< p< \infty . We determine which of them are bounded from H^1(\gamma) to L^1(\gamma) and from L^\infty(v) to BMO( \gamma ). Here H^1(\gamma) and BMO( \gamma ) are the spaces introduced in this setting by the first two authors. Surprisingly, we find that the results depend on the dimension of the ambient space.

Dimension dependent hypercontractivity for Gaussian kernels

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck semigroup driven by a non-diffusive L\'evy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.

Harnack and HWI inequalities on infinite-dimensional spaces

In this paper, the dimensional-free Harnack inequalities are established on infinite-dimensional spaces. More precisely, we establish Harnack inequalities for heat semigroup on based loop group and for Ornstein-Uhlenbeck semigroup on the abstract Wiener space. As an application, we establish the HWI inequality on the abstract Wiener space, which contains three important quantities in one inequality, the relative entropy “H”, Wasserstein distance “W”, and Fisher information “I”.

Harnack Inequalities and Applications for Ornstein–Uhlenbeck Semigroups with Jump

The Harnack inequality established in Rockner and Wang (J Funct Anal 203:237–261, 2003) for generalized Mehler semigroup is improved and generalized. As applications, the log-Harnack inequality, the strong Feller property, the hyper-bounded property, and some heat kernel inequalities are presented for a class of O-U type semigroups with jump. These inequalities and semigroup properties are indeed equivalent, and thus sharp, for the Gaussian case. As an application of the log-Harnack inequality, the HWI inequality is established for the Gaussian case. Perturbations with linear growth are also investigated.

Hypercontractivity on the q-Araki-Woods Algebras

Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the $q$-Ornstein-Uhlenbeck semigroup on the $q$-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the $q$-Araki-Woods algebras. We show that hypercontractivity from $L^p$ to $L^2$ can occur if and only if the generator of the deformation is bounded.

Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces

Dans cet article nous etudions la structure de l'ensemble des points avec densite 1/2 pour les ensemble de perimetre fini dans un espace gaussien infini-dimensionnel. Nous demontrons que, comme dans le cas de dimension finie, la mesure de surface est concentree sur cet ensemble de points. Ici, la definition de points avec densite 1/2 est donnee en utilisant le comportement ponctuel du semigroupe de Ornstein-Uhlembeck.

BV functions in a Hilbert space with respect to a Gaussian measure

Functions of bounded variation in Hilbert spaces endowed with a Gaussian measure γ are studied, mainly in connection with Ornstein-Uhlenbeck semigroups for which γ is invariant.

Fractional Poincaré inequalities for general measures

We prove a fractional version of Poincaré inequalities in the context of R n endowed with a fairly general measure. Namely we prove a control of an L 2 norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein–Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures.

Optimal regularity results in spaces of Hölder continuous functions for some infinite dimensional Ornstein–Uhlenbeck semigroup

We consider the elliptic equation λ φ – Lφ = f where λ > 0 , f is θ -Hölder continuous and L is an Ornstein−Uhlenbeck operator in a Hilbert space H . We show that the mapping D^2φ (with values in the space of Hilbert−Schmidt operators on H ) is θ -Hölder continuous.

Ornstein–Uhlenbeck semi-groups on stratified groups

We consider, in the setting of stratified groups G, two analogues of the Ornstein–Uhlenbeck semi-group, namely Markovian diffusion semi-groups acting on L q ( p d γ ) , whose invariant density p is a heat kernel at time 1 on G. Both generators have the same “carré du champ”. The first semi-group is symmetric on L 2 ( p d γ ) , with generator ∑ i = 1 n X i * X i , where ( X i ) i = 1 n is a basis of the first layer of the Lie algebra of G. The second one is compact on L q ( p d γ ) , 1 < q < ∞ , non-symmetric on L 2 ( p d γ ) and the formal real part of its generator N is ∑ i = 1 n X i * X i . The spectrum of N is the set of non-negative integers if polynomials are dense in L 2 ( p d γ ) , i.e. if G has at most 4 layers; we determine in this case the eigenspaces. When G is step two, we give another description of these eigenspaces, similar to the classical definition of Hermite polynomials by their generating function.

Stokes formula on the Wiener space and n-dimensional Nourdin–Peccati analysis

Extensions of the Nourdin–Peccati analysis to R n -valued random variables are obtained by taking conditional expectation on the Wiener space. Several proof techniques are explored, from infinitesimal geometry, to quasi-sure analysis (including a connection to Stein's lemma), to classical analysis on Wiener space. Partial differential equations for the density of an R n -valued centered random variable Z = ( Z 1 , … , Z n ) are obtained. Of particular importance is the function defined by the conditional expectation given Z of the auxiliary random matrix ( − D L − 1 Z i | D Z j ) , i , j = 1 , 2 , … , n , where D and L are respectively the derivative operator and the generator of the Ornstein–Uhlenbeck semigroup on Wiener space.

Stein’s lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent

We consider a random variable X satisfying almost-sure conditions involving G : = 〈 D X , − D L − 1 X 〉 where D X is X ’s Malliavin derivative and L − 1 is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. A lower- (resp. upper-) bound condition on G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail P [ X > z ] . Bounds of other natures are also given. A key ingredient is the use of Stein’s lemma, including the explicit form of the solution of Stein’s equation relative to the function 1 x > z , and its relation to G . Another set of comparable results is established, without the use of Stein’s lemma, using instead a formula for the density of a random variable based on G , recently devised by the author and Ivan Nourdin. As an application, via a Mehler-type formula for G , we show that the Brownian polymer in a Gaussian environment, which is white-noise in time and positively correlated in space, has deviations of Gaussian type and a fluctuation exponent χ = 1 / 2 . We also show this exponent remains 1 / 2 after a non-linear transformation of the polymer’s Hamiltonian.

A parabolic regularization property of p-logarithmic Sobolev generators

Let N be a Riemannian manifold, M ⊂ N be a domain with smooth boundary, μ a positive measure on M such that M has unit μ-volume. Consider the evolution driven by the p-Laplace-type operator (p > 2) associated to the natural p-energy functional E (p) con- structed from μ, homogeneous Dirichlet boundary conditions on ∂M being assumed. Assume that a single suitable logarithmic inequality holds for E (p) .Then we show that the evolution brings any data belonging to the natural domain of the evolution instantaneously into L q for any q > 2, with quantitative bounds on the Lq norms. (q > 2 arbitrary) for the evolution equation associated to (possibly degenerate or sin- gular) p-Laplacian-like operators on finite volume domains of Riemannian manifolds, Dirichlet boundary conditions being assumed, provided the associated energy func- tional satisfy a single logarithmic Sobolev inequality. This parallels, in the present case, the results discovered by L. Gross in his cel- ebrated paper (11) for the linear case (see also (12) and, without any claim of com- pleteness, the fundamental papers of Federbush, Nelson, Simon and Hoegh-Krohn(9), (13), (15)), but shows a substantial and unexpected difference with that situation, in which it is well known that no more than a L 2 -L p(t) regularization holds, with p(t) smooth and increasing, p(0 )= 2, p(t) → +∞ as t → +∞. The Ornstein-Uhlenbeck semigroup shows the sharpness of that result in the linear case, this being particularly evident in the fact that the eigenfunctions of such operator are unbounded. To start with we shall introduce our setting and the corresponding notation. We consider a connected, smooth Riemannian manifold (N,g) of dimension n endowed with the associated Riemannian measure m .L etM ⊂ N be an open domain with smooth boundary and consider a measurable function V on M. It will assumed here- after, and will be crucial in what follows, that e V is a probabilitymeasure on M ,s o that the μ-volume of M is one, where we set dμ := e V dm .A ll L p spaces and norms will

Density Formula and Concentration Inequalities with Malliavin Calculus

We show how to use the Malliavin calculus to obtain a new exact formula for the density $\rho$ of the law of any random variable $Z$ which is measurable and differentiable with respect to a given isonormal Gaussian process. The main advantage of this formula is that it does not refer to the divergence operator $\delta$ (dual of the Malliavin derivative $D$). The formula is based on an auxilliary random variable $G:= < DZ,-DL^{-1}Z >_H$, where $L$ is the generator of the so-called Ornstein-Uhlenbeck semigroup. The use of $G$ was first discovered by Nourdin and Peccati (PTRF 145 75-118 2009 <a href="http://www.ams.org/mathscinet-getitem?mr=2520122">MR-2520122</a>), in the context of rates of convergence in law. Here, thanks to $G$, density lower bounds can be obtained in some instances. Among several examples, we provide an application to the (centered) maximum of a general Gaussian process. We also explain how to derive concentration inequalities for $Z$ in our framework.