共查询到20条相似文献,搜索用时 31 毫秒
1.
按照Ornstein-Uhlenbeck的思想方法,用Ornstein-Uhlenbeck半群和Ornstein-Uhlenbeck算子的一些重要性质,对Brascamp-Lieb不等式、高斯对数Sobolev不等式、逆Bobkov等周不等式等几个重要的几何与分析不等式给出了另一证明. 相似文献
2.
Abdelhadi Es-Sarhir 《Journal of Differential Equations》2007,233(1):87-104
This paper deals with perturbations of the Ornstein-Uhlenbeck operator on L2-spaces with respect to a Gaussian measure μ. We perturb the generator of the Ornstein-Uhlenbeck semigroup by a certain unbounded, non-linear drift, and show various properties of the perturbed semigroup such as compactness and positivity. Strong Feller property, existence and uniqueness of an invariant measure are discussed as well. 相似文献
3.
We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with
respect to its invariant measure μ. For a reversible process the domain of its generator in L
p
(μ) is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck
operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study
the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup (R
t
) to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator.
We show also that the existence of spectral gap implies a smoothing property of R
t
and provide an estimate for the (appropriately defined) gradient of R
t
φ. Finally, in the Hilbert-Schmidt case, we show tha t for any the function R
t
φ is an (almost) classical solution of a version of the Kolmogorov equation.
Received: 17 September 2001 / Revised version: 3 June 2002 / Published online: 30 September 2002
This work was partially supported by the Small ARC Grant Scheme.
Mathematics Subject Classification (2000): Primary: 60H15, 47F05; Secondary: 60J60, 35R15, 35K15
Key words or phrases: Ornstein-Uhlenbeck operator – Second quantization – Reversibility – Spectral gap – Sobolev spaces – Domain of generator 相似文献
4.
José García-Cuerva Giancarlo Mauceri Peter Sjögren José Luis Torrea 《Journal d'Analyse Mathématique》1999,78(1):281-305
The setting of this paper is Euclidean space with the Gaussian measure. We letL be the associated Laplacian, by means of which the Ornstein-Uhlenbeck semigroup is defined. The main result is a multiplier
theorem, saying that a function ofL which is of Laplace transform type defines an operator of weak type (1,1) for the Gaussian measure. The (distribution) kernel
of this operator is determined, in terms of an integral involving the kernel of the Ornstein-Uhlenbeck semigroup. This applies
in particular to the imaginary powers ofL. It is also verified that the weak type constant of these powers increases exponentially with the absolute value of the exponent.
The four authors have received support from the European Commission via the TMR network “Harmonic Analysis”. The first and
last authors were also partially supported by the Spanish DGICYT, under grant PB97-0030. 相似文献
5.
In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser in [16]. As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then only to be verified on some fixed operator core of L. Furthermore, the setting is also suitable for covering existence and construction problems as required in many applications. The methods are applicable to various, different, Kolmogorov backward evolution problems. As a main part, we apply the extended framework to the (degenerate) spherical velocity Langevin equation. This equation e.g. also appears in applied mathematics as the so-called fiber lay-down process. For the construction of the strongly continuous contraction semigroup we make use of modern hypoellipticity tools and perturbation theory. 相似文献
6.
Luigi Manca 《Potential Analysis》2010,32(1):67-99
We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding
transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal
generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation.
In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation. 相似文献
7.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1997,324(11):1227-1230
We consider a semigroup of Markovian and symmetric operators to which we associate fractional Sobolev spaces Dαp (0 < α < 1 and 1 < p < ∞) defined as domains of fractional powers (−Ap)α/2, where Ap is the generator of the semigroup in Lp. We show under rather general assumptions that Lipschitz continuous functions operate by composition on Dαp if p ≥ 2. This holds in particular in the case of the Ornstein-Uhlenbeck semigroup on an abstract Wiener space. 相似文献
8.
Luigi Manca 《Journal of Evolution Equations》2008,8(2):231-262
We consider a semigroup of operators in the Banach space C
b
(H) of uniformly continuous and bounded functions on a separable Hilbert space H. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply
the result to the transition semigroup and the Kolmogorov operator corresponding to a stochastic PDE in H. For this purpose, we characterize the generator of the transition semigroup on a core.
相似文献
9.
Given a Hilbert space H with a Borel probability measure , we prove the m-dissipativity in L
1(H, ) of a Kolmogorov operator K that is a perturbation, not necessarily of gradient type, of an Ornstein-Uhlenbeck operator. 相似文献
10.
We deal with the least squares estimator for the drift parameters of an Ornstein-Uhlenbeck process with periodic mean function driven by fractional Lévy process. For this estimator, we obtain consistency and the asymptotic distribution. Compared with fractional Ornstein-Uhlenbeck and Ornstein-Uhlenbeck driven by Lévy process, they can be regarded both as a Lévy generalization of fractional Brownian motion and a fractional generaliza- tion of Lévy process. 相似文献
11.
In this short survey we present the application of the Askey-scheme of orthogonal polynomials to define several discrete and continuous distribution types in frame-work of T. Hidaʼs white noise analysis. The results are applied to define fractional versions of the distributions and to solve stochastic differential equations involving the Malliavin derivative, Skorokhod integral and Ornstein-Uhlenbeck operator. 相似文献
12.
本文利用Riemann-Liouville分数积分算子的半群性质以及分数Lévy过程的Wie-ner积分,给出由同一平方可积Lévy过程定义的不同分数Lévy过程之间的积分变换公式. 相似文献
13.
We discuss the solvability of integral equations associated with initial value problems for a nonlinear differential equation of fractional order. The differential operator is the Caputo fractional derivative and the inhomogeneous term depends on the fractional derivative of lower orders. We obtain the existence of at least one solution for integral equations using the Leray–Schauder Nonlinear Alternative for several types of initial value problems. In addition, using the Banach contraction principle, we establish sufficient conditions for unique solutions. Our approach in obtaining integral equations is the “reduction” of the fractional order of the integro-differential equations based on certain semigroup properties of the Caputo operator. 相似文献
14.
We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure). Our approach is based on an iteration of the classical Poincaré inequality, as well as on the use of Malliavin operators, of Stein’s method, and of an (integrated) Mehler’s formula, providing a representation of the Ornstein-Uhlenbeck semigroup in terms of thinned Poisson processes. Our estimates only involve first and second order difference operators, and have consequently a clear geometric interpretation. In particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization, and with non-linear functionals of Poisson shot-noise processes. We discuss two examples of stabilizing functionals in great detail: (i) the edge length of the k-nearest neighbour graph, (ii) intrinsic volumes of k-faces of Voronoi tessellations. In all these examples we obtain rates of convergence (in the Kolmogorov and the Wasserstein distance) that one can reasonably conjecture to be optimal, thus significantly improving previous findings in the literature. As a necessary step in our analysis, we also derive new lower bounds for variances of Poisson functionals. 相似文献
15.
Sonsoles Pérez 《Journal of Geometric Analysis》2001,11(3):491-507
In this work we present several theorems which imply the weak type 1 with respect to the Gaussian measure for the so-called
local part of certain operators associated with the Ornstein-Uhlenbeck semigroup. Particular cases of these operators are
the Riesz transforms of any order and the Littlewood-Paley square function. Also, we study general results based on the “size”
of the operator which ensure the strong type 1 <p < ∞of both the local and global parts. 相似文献
16.
In this article, we give necessary and sufficient conditions for the existence of a weak solution of a Kolmogorov equation perturbed by an inverse-square potential. More precisely, using a weighted Hardy's inequality with respect to an invariant measure μ, we show the existence of the semigroup solution of the parabolic problem corresponding to a generalized Ornstein–Uhlenbeck operator perturbed by an inverse-square potential in L 2(? N ,?μ). In the case of the classical Ornstein–Uhlenbeck operator we obtain nonexistence of positive exponentially bounded solutions of the parabolic problem if the coefficient of the inverse-square function is too large. 相似文献
17.
We prove the existence of an invariant measure for the transition semigroup associated with a nonlinear damped stochastic wave equation in Rn of the Klein--Gordon type. The uniqueness of the invariant measure and the structure of the corresponding Kolmogorov operator are also studied. 相似文献
18.
Many phenomena in mathematical physics and in the theory of stochastic processes are recently described through fractional evolution equations. We investigate a general framework for connections between ordinary non-homogeneous equations in Banach spaces and fractional Cauchy problems. When the underlying operator generates a strongly continuous semigroup, it is known, using a subordination argument, that the fractional evolution equation is well posed. In this case, we provide an explicit form of the solution involving special functions, one example being the Airy function. 相似文献
19.
This paper investigates the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. Based on the operator semigroup theory, we transform the time-space fractional reaction-diffusion equation into an abstract evolution equation. The existence and uniqueness of mild solution to the reaction-diffusion equation are obtained by solving the abstract evolution equation. Finally, we verify the Mittag-Leffler-Ulam stabilities of the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. The results in this paper improve and extend some related conclusions to this topic. 相似文献
20.
In a Banach space, we consider a Cauchy type problem with a left Hadamard fractional derivative of order α ∈ (0, 1) and a Cauchy problem with a regularized Hadamard fractional derivative. We prove the well-posed solvability of such
problems with a bounded operator as well as with the generator of a strongly continuous semigroup. For inverse coefficient
problems, we indicate sufficient conditions for their unique solvability. 相似文献