High-frequency propagation for the Schrödinger equation on the torus

The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schrödinger equation on the standard d-dimensional torus Td. From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigner distributions of the initial data. We characterize them in terms of a new object, the resonant Wigner distribution, which describes high-frequency effects associated to the fraction of the energy of the sequence of initial data that concentrates around the set of resonant frequencies in phase-space T^*Td. This construction is related to that of the so-called two-microlocal semiclassical measures. We prove that any limit μ of the Wigner distributions corresponding to solutions to the Schrödinger equation on the torus is completely determined by the limits of both the Wigner distribution and the resonant Wigner distribution of the initial data; moreover, μ follows a propagation law described by a family of density-matrix Schrödinger equations on the periodic geodesics of Td. Finally, we present some connections with the study of the dispersive behavior of the Schrödinger flow (in particular, with Strichartz estimates). Among these, we show that the limits of sequences of position densities of solutions to the Schrödinger equation on T² are absolutely continuous with respect to the Lebesgue measure.     

Radiative transport limit for the random Schrödinger equation with long-range correlations

In this paper we study the asymptotic phase space energy distribution of solution of the Schrödinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. We show that the Wigner transform of a solution of the random Schrödinger equation converges in probability to the solution of a radiative transfer equation. Moreover, we show that this radiative transfer equation with long-range coupling has a regularizing effect on its solutions. Finally, we give an approximation of this equation in term of a fractional Laplacian. The derivations of these results are based on an asymptotic analysis using perturbed-test-functions, martingale techniques, and probabilistic representations.     

Construction d'un modèle M1-multigroupe pour les équations du transfert radiatifConstruction of a multigroup M1 model for the radiative transfer equations

Though not requiring the great accuracy of a kinetic method, several applications of radiative transfer need to take into account quantities which are not constant over all the frequencies. We introduce a moments model which aims at extending the possibilities of the M1 model proposed by B. Dubroca and J.L. Feugeas [2] in order to solve such applications and keeps the hyperbolicity and physical consistency. To cite this article: R. Turpault, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 331–336.     

Polylogarithmes multiples uniformes en une variable

Various single-valued versions of ordinary polylogarithms Li_n(z) have been constructed by Ramakrishnan, Wojtkowiak, Zagier, and others. These single-valued functions are generalisations of the Bloch–Wigner dilogarithm and have many applications in mathematics. In this Note we show how to construct explicit single-valued versions of multiple polylogarithms in one variable. We prove the functions thus constructed are linearly independent, that they satisfy the shuffle relations, and that every possible single-valued version of multiple polylogarithms in one variable can be obtained in this way. To cite this article: F.C.S. Brown, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Smoothness of Wigner densities on the affine algebra

The non-commutative Malliavin calculus on the Heisenberg–Weyl algebra (see (i) C. R. Acad. Sci. Paris, Sér. I 328 (11) (1999) 1061–1066, (ii) Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (1) (2001) 11–38) is extended to the affine algebra. A differential calculus is established, which generalizes the corresponding commutative integration by parts formulas. As an application we obtain sufficient conditions for the smoothness of Wigner type laws of non-commutative random variables with gamma and continuous binomial marginals. To cite this article: U. Franz et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Ambiguity functions, Wigner distributions and Cohen's class for LCA groups

In this paper we construct a general class of time-frequency representations for LCA groups which parallel Cohen's class for the real line. For this, we generalize the notion of ambiguity function and Wigner distribution to the setting of general LCA groups in such a way that the Plancherel transform of the ambiguity function coincides with the Wigner distribution. Furthermore, properties of the general ambiguity function and Wigner distribution are studied. In detail we characterize those groups whose ambiguity functions and Wigner distributions vanish at infinity or are square-integrable. Finally, we explicitly construct Cohen's class for the group of p-adic numbers, p prime.     

The algebra of harmonic functions for a matrix-valued transfer operator

We analyze matrix-valued transfer operators. We prove that the fixed points of transfer operators form a finite-dimensional C^∗-algebra. For matrix weights satisfying a low-pass condition we identify the minimal projections in this algebra as correlations of scaling functions, i.e., limits of cascade algorithms.     

Hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves

We consider a class of kinetic equations, equipped with a single conservation law, which generate -contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the ``dissipative' sense, equivalent to the notion of entropy solutions for conservation laws, which permits the use of the perturbed test function method and allows for simple proofs. A general compactness framework is obtained for the diffusive scaling in . The radiative transport equations, satisfied by the Wigner function for random acoustic waves, present such a kinetic model that is endowed with conservation of energy. The general theory is used to validate the diffusive approximation of the radiative transport equation.

     

Scaling-sharp dispersive estimates for the Korteweg–de Vries group

We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates is in the spirit of that used to prove small data scattering for the generalized KdV equations. To cite this article: R. Côte, L. Vega, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Asymptotic preserving numerical schemes for a non‐classical radiation transport model for atmospheric clouds

We present numerical schemes for the P1‐moment and M1‐moment approximations of a non‐classical transport equation modeling radiative transfer in atmospheric clouds. In contrast to classical radiative transfer, the photon path‐length is introduced as an additional variable and serves as pseudo‐time in this model. Because clouds may have optically thick regions, we introduce a diffusive scaling and show that the diffusion limits of the moment models and the original equations agree. Furthermore, we show that the numerical schemes also preserve the diffusion asymptotics as well as the set of admissible and realizable states, both for the explicit and the implicit discretization of the pseudo‐time variable. A source iteration‐like method is proposed, and we observe that it converges slowly in the optical thick case, but a suitable initialization can help to overcome this problem. We validate our method in 1D and present simulation results in the 2D‐case for real cloud data. Copyright © 2013 John Wiley & Sons, Ltd.     

Sur l'existence des opérateurs d'onde pour l'équation de Wigner dans les espaces L2,pExistence of wave operators for the Wigner equation in L2,p spaces

Basing on the formalism established by Markovich, we show the completeness of wave operators for the Wigner equation in L². In the second part, using estimations proved by Castella and Perthame on the one hand, and the L^p→L^q estimations for the Schrödinger group on the other hand, we prove the existence of the wave operators in L^2,p spaces. To cite this article: H. Emamirad, P. Rogeon, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 811–816.     

On Euler products and multi-variate Gaussians

In this Note, we extend a recent result of A. Selberg concerning the asymptotic value distribution of Euler products to a multi-dimensional setting. Under certain conditions, an asymptotic development of Edgeworth type is found. To cite this article: D.A. Hejhal, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Testing of the homogeneity of marginal distributions in copula models

A test for the equality of marginal distributions of bi-dimensional distribution functions represented by a parametric copula and completely unknown marginals is proposed. To cite this article: V. Bagdonavičius et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Sur quelques limites de la physique des particules chargées vers la (magnéto)hydrodynamiqueOn some limits in charged particle physics towards (magneto)hydrodynamic equations

We discuss the connection between different scalings limits of the quantum-relativistic Dirac–Maxwell system. In particular we give rigorous results for the quasi-neutral/non-relativistic limit of the Vlasov–Maxwell system: we obtain a magneto-hydro-dynamic system when we consider the magnetic field as a non-relativistic effect and we obtain the Euler equation when we see it as a relativistic effect. A mathematical key is the modulated energy method. To cite this article: Y. Brenier et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 239–244.     

Bounded influence estimators for multivariate lognormal distributions

In this paper we consider the problem of robust estimation of some parameters related to a multivariate lognormal distribution. In this sense, we construct a class of estimators and discuss some of its properties, such as Fisher consistency, robustness and asymptotic normality. To cite this article: A. Toma, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Une approche formelle unifiée des théories de plaques et poutres linéairement élastiques

In this Note we present a formal scaling method that allows for the deduction from three-dimensional linearized elasticity of the equations of shearable structures such as Reissner–Mindlin's equations for plates and Timoshenko's equations for rods, as well as other models of thin structures. This method is based on the requirement that a scaled energy functional possibly including second-gradient terms stay bounded in the limit of vanishing ‘thinness’. To cite this article: B. Miara, P. Podio-Guidugli, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Transfert lisse d'intégrales de KloostermanSmooth transfer of Kloosterman integrals

We establish the existence of smooth transfer between absolute Kloosterman integrals and Kloosterman integrals relative to a quadratic extension. To cite this article: H. Jacquet, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 229–232.     

Numerical error for SDE: Asymptotic expansion and hyperdistributions

The principal part of the error in the Euler scheme for an SDE with smooth coefficients can be expressed as a generalized Watanabe distribution on Wiener space. To cite this article: P. Malliavin, A. Thalmaier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Un algorithme de Schur pour les fonctions de transfert des systèmes surdéterminés invariants dans une direction

We define an analogue of the Schur algorithm for transfer functions of lossless 2D systems which are invariant with respect to one of the variables. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Asymptotic distribution of a Pickands-type estimator of the extreme-value index

One of the main goals of extreme-value analysis is to estimate the probability of rare events given a sample from an unknown distribution. The upper tail behavior of this distribution is described by the extreme-value index ξ. The aim of this Note is to establish the asymptotic distribution of the estimator of $\xi \in \mathbb{R}$ introduced in Gardes and Girard [A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004]. We also give its rate of convergence in some typical situations. To cite this article: L. Gardes, S. Girard, C. R. Acad. Sci. Paris, Ser. I 341 (2005).