Some remarks on degenerate hypoelliptic Ornstein-Uhlenbeck operators

37 pages, 3 figuresInternational audienceWe study degenerate hypoelliptic Ornstein-Uhlenbeck operators in $L^2$ spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be directly recovered from the general analysis of quadratic operators with zero singular spaces. We also provide new resolvent estimates for hypoelliptic Ornstein-Uhlenbeck operators. We show in particular that the spectrum of these non-selfadjoint operators may be very unstable under small perturbations and that their resolvents can blow-up in norm far away from their spectra. Furthermore, we establish sharp resolvent estimates in specific regions of the resolvent set which enable us to prove exponential return to equilibrium

Nonlinear Instability in a Semiclassical Problem

We consider a nonlinear evolution problem with an asymptotic parameter and construct examples in which the linearized operator has spectrum uniformly bounded away from Re z >= 0 (that is, the problem is spectrally stable), yet the nonlinear evolution blows up in short times for arbitrarily small initial data. We interpret the results in terms of semiclassical pseudospectrum of the linearized operator: despite having the spectrum in Re z < -c < 0, the resolvent of the linearized operator grows very quickly in parts of the region Re z > 0. We also illustrate the results numerically

Asymptotic analysis for the generalized langevin equation

Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in \cite{EPR99}. Geometric ergodicity, a homogenization theorem (invariance principle), short time asymptotics and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity \cite{Vil04HPI} is made.Comment: 27 pages, no figures. Submitted to Nonlinearity

SUBELLIPTIC ESTIMATES FOR OVERDETERMINED SYSTEMS OF QUADRATIC DIFFERENTIAL OPERATORS

We prove global subelliptic estimates for systems of quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. In a previous work, we pointed out the existence of a particular linear subvector space in the phase space intrinsically associated to their Weyl symbols, called singular space, which rules a number of fairly general properties of non-elliptic quadratic operators. About the subelliptic properties of these operators, we established that quadratic operators with zero singular spaces fulfill global subelliptic estimates with a loss of derivatives depending on certain algebraic properties of the Hamilton maps associated to their Weyl symbols. The purpose of the present work is to prove similar global subelliptic estimates for overdetermined systems of quadratic operators. We establish here a simple criterion for the subellipticity of these systems giving an explicit measure of the loss of derivatives and highlighting the non-trivial interactions played by the different operators composing those systems. 1

Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with nonnegative real part. We point out that the singular space associated with the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated with the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times. © 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei

Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with nonnegative real part. We point out that the singular space associated with the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated with the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times. © 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei