Ornstein-Uhlenbeck Semigroup Research Articles

Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

AbstractWe consider the spaces $${\text {L}}^p(X,\nu ;V)$$ L p ( X , ν ; V ) , where X is a separable Banach space, $$\mu $$ μ is a centred non-degenerate Gaussian measure, $$\nu :=Ke^{-U}\mu $$ ν : = K e - U μ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions $$F\in W^{1,p}(X,\nu ;V)$$ F ∈ W 1 , p ( X , ν ; V ) , which allows us to show that for every $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and every $$k\in \mathbb {N}$$ k ∈ N the norm in $$W^{k,p}(X,\nu )$$ W k , p ( X , ν ) is equivalent to the graph norm of $$D_H^{k}$$ D H k (the k-th Malliavin derivative) in $${\text {L}}^p(X,\nu )$$ L p ( X , ν ) . To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup $$(T^V(t))_{t\ge 0}$$ ( T V ( t ) ) t ≥ 0 , defined in Section 2.6, as t goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $$(T(t))_{t\ge 0}$$ ( T ( t ) ) t ≥ 0 , and pointwise estimates for $$|D_HT(t)f|_H^p$$ | D H T ( t ) f | H p by means of both $$T(t)|D_Hf|^p_H$$ T ( t ) | D H f | H p and $$T(t)|f|^p$$ T ( t ) | f | p .

On the variation operator for the Ornstein–Uhlenbeck semigroup in dimension one

Consider the variation seminorm of the Ornstein–Uhlenbeck semigroup H_t in dimension one, taken with respect to t. We show that this seminorm defines an operator of weak type (1, 1) for the relevant Gaussian measure. The analogous L^p estimates for 1<p<infty were already known.

On Exponential Splitting Methods for Semilinear Abstract Cauchy problems

Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than frac{pi }{2} (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the semigroup and on the semilinearity) are inspired by the approach of Kato (Math Z 187(4):471–480, 1984) to the local solvability of the Navier–Stokes equation, where the textrm{L}^p - textrm{L}^r-smoothing of the Stokes semigroup was fundamental. The present abstract operator theoretic result is applicable for this latter problem (as was already the result of Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies.

Maximal Operators, 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 -lambda (I+R), being lambda >0 and R a skew-adjoint matrix. The semigroups associated with these Ornstein–Uhlenbeck operators are the basic building blocks of all the normal Ornstein–Uhlenbeck semigroups.

Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups

We consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e. for every initial value, estimating the state at a final time T>0 by taking into account the orbit of the initial value under the semigroup for tin [0,T], measured in a suitable norm. We state a sufficient criterion based on an uncertainty relation and a dissipation estimate and provide two examples of bi-continuous semigroups which share a final state observability estimate, namely the Gauß–Weierstraß semigroup and the Ornstein–Uhlenbeck semigroup on the space of bounded continuous functions. Moreover, we generalise the duality between cost-uniform approximate null-controllability and final state observability estimates to the setting of locally convex spaces for the case of bounded and continuous control functions, which seems to be new even for the case of Banach spaces.

De Bruijn identities in different Markovian channels

De Bruijn's identity in information theory states that if u is the solution of the heat equation, then the time derivative of the Shannon entropy for this solution is equal to the amount of Fisher information at u. In this article, we show how this identity changes if we replace the heat channel by the Fokker Planck, or passing from Fokker Planck to Ornstein-Uhlenbeck channels. Through these passages we investigate the different properties of these solutions. We exclusively dissect different properties of Ornstein-Uhlenbeck semigroup given by the Mehler formula expression. For more information see https://ejde.math.txstate.edu/Volumes/2023/12/abstr.html

Estimates for the Heat Flow in Optimal Spaces of Unbounded Initial Data in mathbb {R}^{{d}} and Applications to the Ornstein–Uhlenbeck Semigroup

In this paper, we improve some estimates from [7] for solutions of the heat equation with initial conditions that are large ‘at infinity’ and employ them to obtain suitable estimates on the solutions of the Ornstein–Uhlenbeck semigroup [solutions of u_t-Delta u=alpha x cdot nabla u] for the same class of unbounded data.

Generalized Ornstein–Uhlenbeck semigroups in weighted Lp-spaces on Riemannian manifolds

Let E be a Hermitian vector bundle over a Riemannian manifold M with metric g, and let ∇ be a metric covariant derivative on E. We study the generalized Ornstein–Uhlenbeck differential expression P∇=∇†∇u+∇(dϕ)♯u−∇Xu+Vu, where ∇† is the formal adjoint of ∇, (dϕ)♯ is the vector field corresponding to dϕ via g, X is a smooth real vector field on M, and V is a self-adjoint locally integrable section of the endomorphsim bundle EndE. In the setting of a geodesically complete M, we establish a sufficient condition for the equality of the maximal and minimal realizations of P∇ in the (weighted) space ΓLμp(E) of Lμp-type sections of E, where 1<p<∞ and dμ=e−ϕdνg, with νg being the usual volume measure. Furthermore, we show that (the negative of) the maximal realization −Hp,max generates an analytic quasi-contractive semigroup in ΓLμp(E), 1<p<∞. Additionally, in the same context, we establish a coercivity estimate, which leads to the so-called separation property of the covariant Schrödinger operator (that is, P∇ with X≡0 and ϕ≡0) in the (unweighted) space ΓLp(E), 1<p<∞. With the generation result at our disposal, we describe a Feynman–Kac representation for the Lμp-semigroup generated by Hp,max, 1<p<∞. For the Ornstein–Uhlenbeck differential expression acting on functions, that is, Pd=Δu+(dϕ)♯u−Xu+Vu, where Δ is the (non-negative) scalar Laplacian on M and V is a locally integrable real-valued function, we consider another way of realizing Pd as an operator in Lμp(M) and, by imposing certain geometric conditions on M, we prove another semigroup generation result. The study of the mentioned realization of Pd depends, among other things, on the fulfillment of the so-called Lp-Calderón–Zygmund inequality on M.

Two generalizations of Mehler's formula in white noise analysis

ABSTRACT Mehler's formula is an important tool in Gaussian analysis. In this article, we study two generalizations of Mehler's formula for the Ornstein–Uhlenbeck semigroup, i.e. the semigroup generated by the number operator. The first generalization leads to transformation groups which have as infinitesimal generator a perturbation of the number operator with suitable integral kernel operators, which are well studied in white noise analysis. For the second one, we characterize the complex Hida measures for which a version of Mehler's formula for the Ornstein–Uhlenbeck semigroup can be extended to. We apply this result to the Feynman integrand for a quadratic potential. Here the time independent eigenstates of the considered transformation groups and the time evolution of eigenvalues are provided.

New Gaussian Riesz transforms on variable Lebesgue spaces

We give sufficient conditions on the exponent p: ℝd → [1, ∞) for the boundedness of the non-centered Gaussian maximal function on variable Lebesgue spaces Lp(·) (γd), as well as of the new higher order Riesz transforms associated with the Ornstein—Uhlenbeck semigroup, which are the natural extensions of the supplementary first order Gaussian Riesz transforms defined by A. Nowak and K. Stempak in [25].

On the maximal operator of a general Ornstein–Uhlenbeck semigroup

AbstractIf Q is a real, symmetric and positive definite $$n\times n$$ n × n matrix, and B a real $$n\times n$$ n × n matrix whose eigenvalues have negative real parts, we consider the Ornstein–Uhlenbeck semigroup on $$\mathbb {R}^n$$ R n with covariance Q and drift matrix B. Our main result says that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure. The proof has a geometric gist and hinges on the “forbidden zones method” previously introduced by the third author.

Ornstein–Uhlenbeck semigroups on star graphs

&lt;p style='text-indent:20px;'&gt;We prove first existence of a classical solution to a class of parabolic problems with unbounded coefficients on metric star graphs subject to Kirchhoff-type conditions. The result is applied to the Ornstein–Uhlenbeck and the harmonic oscillator operators on metric star graphs. We give an explicit formula for the associated Ornstein–Uhlenbeck semigroup and give the unique associated invariant measure. We show that this semigroup inherits the regularity properties of the classical Ornstein–Uhlenbeck semigroup on &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbb{R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;

Characterizations of sets of finite perimeter using the Ornstein-Uhlenbeck semigroup in the Gauss space

The goal of this paper is to establish the relation between the notion of sets of finite perimeter and the theory of the Ornstein-Uhlenbeck semigroup in the setting of the Gauss space. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the Ornstein-Uhlenbeck semigroup and we also give a new characterization of BV functions in terms of a near-diagonal energy in this space.

Boundedness of the Maximal Function of the Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces with respect to the Gaussian measure and consequences

The main result of this paper is the proof of the boundedness of the Maximal Function T* of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd, on Gaussian variable Lebesgue spaces Lp(.) (γd); under a condition of regularity on p(.) following [5] and [8]. As an immediate consequence of that result, the Lp(.) (γd)-boundedness of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd is obtained. Another consequence of that result is the Lp(.) (γd)-boundedness of the Poisson-Hermite semigroup and the Lp(.) (γd)- boundedness of the Gaussian Bessel potentials of order β &gt; 0.

Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $\mathbb {L}^{1}$

It is well known that some important Markov semi-groups have a "regularization effect" -- as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in $L^p$ for $p>1$. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in $\mathbb{R}^n$, as well as for the $M/M/\infty$ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.

Ornstein-Uhlenbeck Semigroup on the dual space of Gelfand-Shilov Spaces of Beurling type

We use a previously obtained topological characterization of Gelfand-Shilov spaces of Beurling type to characterize its dual using Riesz representation theorem. Using the characterization of the dual space equipped with the weak topology, we study the action of Ornstein-Uhlenbeck Semigroup on the dual space.

The maximal operator of a normal Ornstein-Uhlenbeck semigroup is of weak type (1,1)

Consider a normal Ornstein--Uhlenbeck semigroup in $\Bbb{R}^n$, whose covariance is given by a positive definite matrix. The drift matrix is assumed to have eigenvalues only in the left half-plane. We prove that the associated maximal operator is of weak type $(1,1)$ with respect to the invariant measure. This extends earlier work by G. Mauceri and L. Noselli. The proof goes via the special case where the matrix defining the covariance is $I$ and the drift matrix is diagonal.

On interweaving relations

Interweaving relations are introduced and studied here in a general Markovian setting as a strengthening of usual intertwining relations between semigroups, obtained by adding a randomized delay feature. They provide a new classification scheme of the set of Markovian semigroups which enables to transfer from a reference semigroup and up to an independent warm-up time, some ergodic, analytical and mixing properties including the φ-entropy convergence to equilibrium, the hyperboundedness and, when the warm-up time is deterministic, the cut-off phenomena. We also present several useful transformations that preserve interweaving relations. We provide a variety of examples of interweaving relations ranging from classical, discrete, and non-local Laguerre and Jacobi semigroups to degenerate hypoelliptic Ornstein-Uhlenbeck semigroups and some non-colliding particle systems.

Ornstein-Uhlenbeck semigroups in infinite dimension.

This is a survey paper about Ornstein-Uhlenbeck semigroups in infinite dimension and their generators. We start from the classical Ornstein-Uhlenbeck semigroup on Wiener spaces and then discuss the general case in Hilbert spaces. Finally, we present some results for Ornstein-Uhlenbeck semigroups on Banach spaces. This article is part of the theme issue 'Semigroup applications everywhere'.

The Ornstein-Uhlenbeck semigroup in finite dimension.

We gather the main known results concerning the non-degenerate Ornstein-Uhlenbeck semigroup in finite dimension. This article is part of the theme issue 'Semigroup applications everywhere'.