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

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-…

On the Ornstein-Uhlenbeck operator in convex sets of Banach spaces

We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space $X$. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincaré inequalities, and thanks to these inequalities we deduce the spectral properties of the Ornstein-Uhlenbeck operator.

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.

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.

Convergence to equilibrium for hypoelliptic non-symmetric Ornstein-Uhlenbeck type operators

We study a generalized curvature dimension inequality which is suitable for subelliptic Ornstein-Uhlenbeck type operators and deduce convergence to equilibrium in the $L^2$ and entropic sense. The main difficulty is that the operators we consider may not be symmetric. Our results apply in particular to Ornstein-Uhlenbeck operators on two-step Carnot groups.

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.

$L^p$-$L^q$ 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…

L^2-Theory for non-symmetric Ornstein-Uhlenbeck semigroups on domains

We present some new results on analytic Ornstein-Uhlenbeck semigroups and use them to extend recent work of Da Prato and Lunardi for Ornstein-Uhlenbeck semigroups on open domains O to the non-symmetric case. Denoting the generator of the semigroup by L_O, we obtain sufficient conditions in order that the domain Dom(\sqrt{-L_O}) be a first order Sobolev space.

A spectral mapping theorem for perturbed Ornstein-Uhlenbeck operators on L^2(R^d)

We consider Ornstein-Uhlenbeck operators perturbed by a radial potential. Under weak assumptions we prove a spectral mapping theorem for the generated semigroup. The proof relies on a perturbative construction of the resolvent, based on angular separation, and the Gearhart-Prüß Theorem.

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.