Endpoint Results for the Riesz Transform of the Ornstein–Uhlenbeck Operator

Journal of Fourier Analysis and Applications - In this paper we introduce a new atomic Hardy space $$X^1(gamma )$$ adapted to the Gauss measure $$gamma $$ , and prove the boundedness of the first...     

Semigroup and Riesz transform for the Dunkl–Schrödinger operators

Semigroup Forum - Let $$L_k=-\Delta _k+V$$ be the Dunkl–Schrödinger operators, where $$\Delta _k=\sum _{j=1}^dT_j^2$$ is the Dunkl Laplace operator associated to the Dunkl operators...     

Global L p estimates for degenerate Ornstein–Uhlenbeck operators

We consider a class of degenerate Ornstein–Uhlenbeck operators in ${\mathbb{R}^{N}}We consider a class of degenerate Ornstein–Uhlenbeck operators in \mathbbR^N{\mathbb{R}^{N}} , of the kind

A o ?i, j=1p0aij?xixj2 + ?i, j=1Nbijxi?xj\mathcal{A}\equiv\sum_{i, j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} + \sum_{i, j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}      4. Ornstein–Uhlenbeck operators with time periodic coefficients      Giuseppe Da Prato  Alessandra Lunardi 《Journal of Evolution Equations》2007,7(4):587-614 We study the realization of the differential operator in the space of continuous time periodic functions, and in L 2 with respect to its (unique) invariant measure. Here L(t) is an Ornstein-Uhlenbeck operator in , such that L(t + T) = L(t) for each .       5. Parameter identification for the Hermite Ornstein–Uhlenbeck process      Assaad  Obayda  Tudor  Ciprian A. 《Statistical Inference for Stochastic Processes》2020,23(2):251-270 Statistical Inference for Stochastic Processes - By using the analysis on Wiener chaos, we study the behavior of the quadratic variations of the Hermite Ornstein–Uhlenbeck process, which is...      6. On Ornstein–Uhlenbeck driven by Ornstein–Uhlenbeck processes      《Statistics & probability letters》2014 We investigate the asymptotic behavior of the maximum likelihood estimators of the unknown parameters of positive recurrent Ornstein–Uhlenbeck processes driven by Ornstein–Uhlenbeck processes.      7. The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup      Giancarlo Mauceri  Luana Noselli 《Journal of Fourier Analysis and Applications》2009,15(2):179-200 Let (ℋ t ) t≥0 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 ∞ the invariant measure for (ℋ t ) t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L 2(γ ∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ* f(x)=sup  t≥o |ℋ t f(x)| is of weak type 1 with respect to the invariant measure γ ∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L p (γ ∞) if and only if 1<p≤∞.       8. Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process      Gehringer  Johann  Li  Xue-Mei 《Journal of Theoretical Probability》2022,35(1):426-456 Journal of Theoretical Probability - We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein–Uhlenbeck process, providing the foundation for the...      9. Sharp large deviations for the non-stationary Ornstein–Uhlenbeck process      Bernard Bercu  Laure Coutin  Nicolas Savy 《Stochastic Processes and their Applications》2012,122(10):3393-3424 For the Ornstein–Uhlenbeck process, the asymptotic behavior of the maximum likelihood estimator of the drift parameter is totally different in the stable, unstable, and explosive cases. Notwithstanding this trichotomy, we investigate sharp large deviation principles for this estimator in the three situations. In the explosive case, we exhibit a very unusual rate function with a shaped flat valley and an abrupt discontinuity point at its minimum.      10. Exact conditions for no ruin for the generalised Ornstein–Uhlenbeck process      Damien Bankovsky  Allan Sly 《Stochastic Processes and their Applications》2009       11. Parameter estimation for fractional Ornstein–Uhlenbeck processes      Yaozhong Hu  David Nualart 《Statistics & probability letters》2010,80(11-12):1030-1038       12. Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup      Raffaella Carbone  Emanuela Sasso 《Probability Theory and Related Fields》2008,140(3-4):505-522 We prove hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup on the entire algebra of bounded operators on a separable Hilbert space h. We exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain. Then we deduce a logarithmic Sobolev inequality for the semigroup and gain an elementary estimate of the best constant.      13. Regularity for semigroups of Ornstein–Uhlenbeck processes      Jian Wang 《Positivity》2013,17(2):205-221 Under mild conditions on the characteristic exponent or the symbol of Lévy process, we derive explicit estimates for L p (dx) → L q (dx) (1 ≤ p ≤ q ≤ ∞) norms of semigroups and their gradients of the associated Lévy driven Ornstein–Uhlenbeck process. Our result efficiently applies to the class of Lévy driven Ornstein–Uhlenbeck processes, where the asymptotic behaviour near infinity for the symbol of Lévy process is known.      14. Hydrodynamic limit for interacting Ornstein–Uhlenbeck particles      《Stochastic Processes and their Applications》2002,102(1):139-158 We consider a system of interacting Ornstein–Uhlenbeck particles moving in a d-dimensional torus. The interaction between particles is given by a short-range superstable pair potential V. We prove that, in a diffusive scaling limit, the density of particles satisfies a non-linear partial differential equation. This generalizes to higher dimensions a result of Olla and Varadhan (cf. (Comm. Math. Phys. 125 (1993) 523)).      15. Exact asymptotic bias for estimators of the Ornstein–Uhlenbeck process      Denis Bosq 《Statistical Inference for Stochastic Processes》2010,13(2):133-145 We study the bias and the bias derivative for a family \({\mathcal{F}}\) of asymptotically efficient estimators of the Ornstein–Uhlenbeck process. That family contains the maximum likelihood, the conditional maximum likelihood and the empirical estimators. We show that, if g(θ T ) is an estimator of g(θ), where θ is the parameter and \({\theta_{T} \in \mathcal{F}}\), then, under mild conditions, $T\,E\left[g(\theta_{T})-g(\theta)\right]\xrightarrow[T\rightarrow\infty]{}c_{\theta}g^{\prime}(\theta)+\theta{g}^{\prime\prime}(\theta),$ where c θ is an explicit constant that only depends on the choice of θ T . In particular, if θ T is one of the three previous estimators, one has $T\,E_{\theta}(\theta_{T}-\theta)\xrightarrow[T\rightarrow\infty]\,2.$       16. Functional calculus for some perturbations of the Ornstein–Uhlenbeck operator      Andrea Carbonaro 《Mathematische Zeitschrift》2009,262(2):313-347 In a recent paper García-Cuerva et al. have shown that for every p in (1,∞) the symmetric finite-dimensional Ornstein–Uhlenbeck operator has a bounded holomorphic functional calculus on L p in the sector of angle . We prove a similar result for some perturbations of the Ornstein–Uhlenbeck operator. Work partially supported by the Progetto Cofinanziato MIUR “Analisi Armonica” and the Gruppo Nazionale INdAM per l’Analisi Matematica, la Probabilitàe le loro Applicazioni.      17. Sharp frequency bounds for eigenfunctions of the Ornstein–Uhlenbeck operator      Tobias?Holck?ColdingEmail author  William?P.?MinicozziII 《Calculus of Variations and Partial Differential Equations》2018,57(5):138 We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein–Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.      18. Hyperbolic Ornstein–Uhlenbeck Process      A. N. Borodin 《Journal of Mathematical Sciences》2016,219(5):631-638       19. Finite speed of propagation and off-diagonal bounds for Ornstein–Uhlenbeck operators in infinite dimensions      Jan van Neerven  Pierre Portal 《Annali di Matematica Pura ed Applicata》2016,195(6):1889-1915       20. On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes      René L. Schilling  Jian Wang 《Journal of Evolution Equations》2012,12(1):119-140 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.

Ornstein–Uhlenbeck operators with time periodic coefficients

We study the realization of the differential operator in the space of continuous time periodic functions, and in L ² with respect to its (unique) invariant measure. Here L(t) is an Ornstein-Uhlenbeck operator in , such that L(t + T) = L(t) for each .      

Parameter identification for the Hermite Ornstein–Uhlenbeck process

Statistical Inference for Stochastic Processes - By using the analysis on Wiener chaos, we study the behavior of the quadratic variations of the Hermite Ornstein–Uhlenbeck process, which is...     

On Ornstein–Uhlenbeck driven by Ornstein–Uhlenbeck processes

We investigate the asymptotic behavior of the maximum likelihood estimators of the unknown parameters of positive recurrent Ornstein–Uhlenbeck processes driven by Ornstein–Uhlenbeck processes.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 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 _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process

Journal of Theoretical Probability - We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein–Uhlenbeck process, providing the foundation for the...     

Sharp large deviations for the non-stationary Ornstein–Uhlenbeck process

For the Ornstein–Uhlenbeck process, the asymptotic behavior of the maximum likelihood estimator of the drift parameter is totally different in the stable, unstable, and explosive cases. Notwithstanding this trichotomy, we investigate sharp large deviation principles for this estimator in the three situations. In the explosive case, we exhibit a very unusual rate function with a shaped flat valley and an abrupt discontinuity point at its minimum.     

Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup

We prove hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup on the entire algebra of bounded operators on a separable Hilbert space h. We exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain. Then we deduce a logarithmic Sobolev inequality for the semigroup and gain an elementary estimate of the best constant.     

Regularity for semigroups of Ornstein–Uhlenbeck processes

Under mild conditions on the characteristic exponent or the symbol of Lévy process, we derive explicit estimates for L ^p (dx) → L ^q (dx) (1 ≤ p ≤ q ≤ ∞) norms of semigroups and their gradients of the associated Lévy driven Ornstein–Uhlenbeck process. Our result efficiently applies to the class of Lévy driven Ornstein–Uhlenbeck processes, where the asymptotic behaviour near infinity for the symbol of Lévy process is known.     

Hydrodynamic limit for interacting Ornstein–Uhlenbeck particles

We consider a system of interacting Ornstein–Uhlenbeck particles moving in a d-dimensional torus. The interaction between particles is given by a short-range superstable pair potential V. We prove that, in a diffusive scaling limit, the density of particles satisfies a non-linear partial differential equation. This generalizes to higher dimensions a result of Olla and Varadhan (cf. (Comm. Math. Phys. 125 (1993) 523)).     

Exact asymptotic bias for estimators of the Ornstein–Uhlenbeck process

We study the bias and the bias derivative for a family \({\mathcal{F}}\) of asymptotically efficient estimators of the Ornstein–Uhlenbeck process. That family contains the maximum likelihood, the conditional maximum likelihood and the empirical estimators. We show that, if g(θ _T ) is an estimator of g(θ), where θ is the parameter and \({\theta_{T} \in \mathcal{F}}\), then, under mild conditions,

$T\,E\left[g(\theta_{T})-g(\theta)\right]\xrightarrow[T\rightarrow\infty]{}c_{\theta}g^{\prime}(\theta)+\theta{g}^{\prime\prime}(\theta),$

where c _θ is an explicit constant that only depends on the choice of θ _T . In particular, if θ _T is one of the three previous estimators, one has

$T\,E_{\theta}(\theta_{T}-\theta)\xrightarrow[T\rightarrow\infty]\,2.$

     

Functional calculus for some perturbations of the Ornstein–Uhlenbeck operator

In a recent paper García-Cuerva et al. have shown that for every p in (1,∞) the symmetric finite-dimensional Ornstein–Uhlenbeck operator has a bounded holomorphic functional calculus on L ^p in the sector of angle . We prove a similar result for some perturbations of the Ornstein–Uhlenbeck operator. Work partially supported by the Progetto Cofinanziato MIUR “Analisi Armonica” and the Gruppo Nazionale INdAM per l’Analisi Matematica, la Probabilitàe le loro Applicazioni.     

Sharp frequency bounds for eigenfunctions of the Ornstein–Uhlenbeck operator

We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein–Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.     

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.