Riesz transforms of a general Ornstein--Uhlenbeck semigroup

关于广义Ornstein-Uhlenbeck半群

On the Generalized Ornstein-Uhlenbeck Semigroup

本文在Dunkl微分-差分算子框架下建立Ornstein-Uhlenbeck半群的若干函数不等式，并由此推导出该半群的Sobolev-对数性质与超压缩性质。

On the Coupling Property and the Liouville Theorem for Ornstein-Uhlenbeck Processes

Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein-Uhlenbeck processes. Then we present the explicit coupling property of Ornstein-Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein-Uhlenbeck processes via the symbol.

Maximal operator, Littlewood-Paley functions and variation operators associated with nonsymmetric Ornstein-Uhlenbeck operators

In this paper we establish $L^p$ boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck operators. We consider the Ornstein-Uhlenbeck operators defined by the identity as the covariance matrix and having a drift given by the matrix $-λ(I+R)$, being $λ>0$ and $R$ a skew-adjoint matrix. The semigroup associated with these Ornstein-Uhlenbeck operators are the basic building blocks of all normal Ornstein-Uhlenbeck semigroups.

Endpoint results for the Riesz transform of the Ornstein-Uhlenbeck operator

In this paper we introduce a new atomic Hardy space $X^1(γ)$ adapted to the Gauss measure $γ$, and prove the boundedness of the first order Riesz transform associated with the Ornstein-Uhlenbeck operator from $X^1(γ)$ to $L^1(γ)$. We also provide a new, short and almost self-contained proof of its weak-type $(1,1)$.

Sharp exponential inequalities for the Ornstein-Uhlenbeck operator

The optimal constants in a class of exponential type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space are detected. The existence of extremal functions in the relevant inequalities is also established. Our results disclose analogies and dissimilarities in comparison with Adams' inequality for the Laplace operator, a companion of our inequalities in the Euclidean space.

Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

We study bounded holomorphic functional calculus for nonsymmetric infinite dimensional Ornstein-Uhlenbeck operators ${\mathscr L}$. We prove that if $-{\mathscr L}$ generates an analytic semigroup on $L^{2}(γ_{\infty})$, then ${\mathscr L}$ has bounded holomorphic functional calculus on $L^{r}(γ_{\infty})$, $1<r<\infty$, in any sector of angle $θ>θ^{*}_{r}$, where $γ_{\infty}$ is the associated invariant measure and $θ^{*}_{r}$ the sectoriality angle of ${\mathscr L}$ on $L^{r}(γ_{\infty})$. The angle $θ^{*}_{r}$ is optimal. In particular our result applies to any nondegenerate finite dimensio…

An L^1-estimate for certain spectral multipliers associated with the Ornstein--Uhlenbeck operator

We study a class of spectral multipliers ϕ(L) for the Ornstein--Uhlenbeck operator L arising from the Gaussian measure on R^n and find a sufficient condition for integrability of ϕ(L)f in terms of the admissible conical square function and a maximal function.

On the eigenfunctions of the complex Ornstein-Uhlenbeck operators

Starting from the 1-dimensional complex-valued Ornstein-Uhlenbeck process, we present two natural ways to imply the associated eigenfunctions of the 2-dimensional normal Ornstein-Uhlenbeck operators in the complex Hilbert space $L_{\Cnum}^2(μ)$. We call the eigenfunctions Hermite-Laguerre-Ito polynomials. In addition, the Mehler summation formula for the complex process are shown.

量子奥本海姆-乌伦贝克半群族的超收缩性

Hypercontractivity for a family of quantum Ornstein-Uhlenbeck semigroups

本文研究了一族量子奥本海姆-乌伦贝克半群的超收缩性，并利用梅克斯纳多项式证明了相关时间最优阶的结果。

Optimal angle of the holomorphic functional calculus for the classical Ornstein-Uhlenbeck operator on $L^p$

We give a simple proof of the fact that the classical Ornstein-Uhlenbeck operator $L$ is R-sectorial of angle $arcsin|1-2/p|$ on $L^{p}(\mathbb{R}^{n},\exp(-|x|^2/2)dx)$ (for $1<p<\infty$). Applying the abstract holomorphic functional calculus theory of Kalton and Weis, this immediately gives a new proof of the fact that $L$ has a bounded $H^{\infty}$ functional calculus with this optimal angle.

The maximal operator associated to a non-symmetric Ornstein-Uhlenbeck semigroup

Let (H_t) be the Ornstein-Uhlenbeck semigroup on R^d with covariance matrix I and drift matrix λ(R-I), where λ>0 and R is a skew-adjoint matrix and denote by γ_\infty the invariant measure for (H_t). Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L^2(γ_\infty). We prove that if the matrix R generates a one-parameter group of periodic rotations then the maximal operator associated to the semigroup is of weak type 1 with respect to the invariant measure. We also prove that the maximal operator associated to an arbitrary normal Ornstein-…

L1-spectrum of Banach space valued Ornstein-Uhlenbeck operators

We characterize the $L^1(E;μ_\infty)$-spectrum of the Ornstein-Uhlenbeck operator, where $μ_\infty$ is the invariant measure for the Ornstein-Uhlenbeck semigroup. The main result covers the general case of an infinite-dimensional Banach space E under the assumption that the point spectrum of drift operator $A^*$ is nonempty. This extends several recent related results.

On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator

We study the orthogonality of the generalized eigenspaces of an Ornstein--Uhlenbeck operator $\mathcal L$ in $\mathbb{R}^N$, with drift given by a real matrix $B$ whose eigenvalues have negative real parts. If $B$ has only one eigenvalue, we prove that any two distinct generalized eigenspaces of $\mathcal L$ are orthogonal with respect to the invariant Gaussian measure. Then we show by means of two examples that if $B$ admits distinct eigenvalues, the generalized eigenspaces of $\mathcal L$ may or may not be orthogonal.

$L^p$-$L^q$ off-diagonal estimates for the Ornstein--Uhlenbeck semigroup: some positive and negative results

We investigate $L^p(γ)$-$L^q(γ)$ off-diagonal estimates for the Ornstein-Uhlenbeck semigroup $(e^{tL})_{t > 0}$. For sufficiently large $t$ (quantified in terms of $p$ and $q$) these estimates hold in an unrestricted sense, while for sufficiently small $t$ they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.

Littlewood-Paley functions associated with general Ornstein-Uhlenbeck semigroups

In this paper we establish $L^p(\mathbb{R}^d,γ_\infty)$-boundedness properties for square functions involving time and spatial derivatives of Ornstein-Uhlenbeck semigroups. Here $γ_\infty$ denotes the invariant measure. In order to prove the strong type results for $1<p<\infty$ we use $R$-boundedness. The weak type (1,1) property is established by studying separately global and local operators defined for the square Littlewood-Paley functions. By the way we prove $L^p(\mathbb{R}^d,γ_\infty)$-boundedness properties for maximal and variation operators for Ornstein-Uhlenbeck semigroups.

On the Jordan decomposition for a class of non-symmetric Ornstein-Uhlenbeck operators

In this paper, we calculate the Jordan decomposition (or say, the Jordan canonical form) for a class of non-symmetric Ornstein-Uhlenbeck operators with the drift coefficient matrix being a Jordan block and the diffusion coefficient matrix being identity multiplying a constant. For the 2-dimensional case, we present all the general eigenfunctions by the induction. For the 3-dimensional case, we divide the calculating of the Jordan decomposition into several steps (the key step is to do the canonical projection onto the homogeneous Hermite polynomials, next we use the theory of systems of linear…

Consistency of the plug-in functional predictor of the Ornstein-Uhlenbeck process in Hilbert and Banach spaces

New results on functional prediction of the Ornstein-Uhlenbeck process in an autoregressive Hilbert-valued and Banach-valued frameworks are derived. Specifically, consistency of the maximum likelihood estimator of the autocorrelation operator, and of the associated plug-in predictor is obtained in both frameworks.