Integration par parties dans l'espace de Wiener et approximation du temps local

Summary Let be a real-valued stochastic process having a continuous local timeL(u,t),u —, 0tT andX (t) = ( *X)(t),t 0, the regularization ofX by means of the convolution with the approximation of unity . The main theorem in this paper (Theorem 3.5) is a generalization of various results about the approximation (for fixedu) of the local timeL(u, ) by means of a convenient normalization of the numberN ^X (u;) of crossings of the processX with the levelu. Especially, this Theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to Azais [A2]). The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the Wiener space.

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Support partill du CICYT, N^o PB86-0238     

Approximation du temps local des surfaces gaussiennes

Summary LetX be a centered stationary Gaussian stochastic process with ad-dimensional parameter (d2),F its spectral measure, (x denotes the Euclidean norm ofx). We consider regularizations of the trajectories ofX by means of convolutions of the formX (t)=( *X)(t) where stands for an approximation of unity (as tends to zero) satisfying certain regularity conditions.The aim of this paper is to recover the local time ofX at a given levelu, as a limit of appropriate normalizations of the geometric measure of theu-level set of the regular approximating processesX . A part of the difficulties comes from the fact that the geometric behavior of the covariance of the Gaussian processX can be a complex one as approaches O.The results are onL ²-convergence and include bounds for the speed of convergence.L ^presults may be obtained in similar ways, but almost sure convergence or simultaneous convergence for the various values ofu do not seem to follow from our methods. In Sect. 3 we have included examples showing a diversity of geometric behaviors, especially in what concerns the dependence on the thickness of the set in which the covariance of the original processX is irregular.Some technical results of analytic nature are included as appendices in Sect. 4.     

Transformation de Fourier et temps d'occupation browniens

Summary In this paper, we study oscillatory stochastic integrals of the form where is a non zero parameter andg a square integrable function. We study integrability properties of () and its behavior as a function of , using stochastic calculus techniques: martingale theory, representation of Itô for a random variable of the Wiener space, lemma of Garsia-Rodemich-Rumsey .... We also obtain limit theorems in law related to the variables () based upon an asymptotic version of a theorem of Knight on orthogonal continuous martingales.We consider the random measure, image by the Brownian motion of the unbounded measure 1_[0,] (s)g(s) ds; we prove the existence and the continuity of an occupation time density.Finally, under a stronger integrability condition ong, we show the existence of a density for the law of (), using Malliavin's calculus.     

Applications des processus de Dirichlet aux temps locaux et temps locaux d'intersection d'un mouvement Brownien

Résumé On montre qu'une large classe d'intégrales associées au mouvement Brownien dans ^d appartient localement à l'espace des processus de Dirichlet faibles. Les transformées de Riesz, les potentiles de Riesz et les puissances fractionnaires du Laplacien des temps locaux d'intersection Browniens sont étudiés. On déduit de nouveaux théorèmes limites pour certaines intégrales doubles dans lesquelles intervient le mouvement Brownien.

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Summary A large class of integrals related to the Brownian motion in ^d is shown to belong locally to the space of weak dirichlet processes. Riesz transforms, Riesz potentials and fractional powers of Laplacean of Brownian local times of intersection are studied. We deduce new limit theorems for certain double integrals concerning the Brownian motion.

     

Inegalités Barlow-Yor pour les temps locaux d'intersection de deux mouvements Browniens plans

Summary We proveL _p> majorations for the suprema of the intersection local time of two planar Brownian motions. We deduce a sufficient condition on the correlation of two Brownian motions for their i.l.t. to be continuous.

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Inegalités Barlow-Yor pour les temps locaux d'intersection de deux mouvements Browniens plans

Résumé On prouve des majorations en normeL _p pour le suprema des temps locaux d'intersection de deux mouvements browniens plans, renormalisés ou non. On en déduit une condition suffisante sur la corrélation entre deux browniens pour que leur t.l.i. soient continus.

     

Limite semi-classique pour l'équation de Schrödinger non-linéaire avec potentiel harmonique

This Note is dedicated to the semiclassical limit of the nonlinear focusing Schrödinger equation with a harmonic potential. The method does not use a linearization argument as is usually done, but the conservation laws (quantum and classical) and the stability of the ground state. To cite this article: S. Keraani, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Grandes déviations du temps local du mouvement brownien fractionnaire

The purpose of this paper is to prove a large deviation principle for a local time of fractional Brownian motion B^H for all H∈(0,1). To cite this article: E.H. Lakhel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 797–801.     

Torseurs pour les motifs et pour les représentationsp-adiques potentiellement de typeCM

Sans résumé     

Transplantation harmonique,transplantation par modules,et théorèmes isopérimétriques

Summary The “harmonic transplantation” allows to extend some isoperimetric theorems, so far proved by conformal mapping, to higher connectivity and to higher dimensions; for the first eigenvalue λ₁ of a membrane, it again can give only upper bounds.—The “transplantation by moduli” is much more flexible; for example, it leads to a simple one-dimensional interpretation of the Rayleigh-Faber-Krahn theorem.      

Solutions globales des équations d'Einstein–Maxwell avec jauge harmonique et jauge de Lorenz

Adapting a method of Lindblad and Rodnianski, we prove existence of global solutions for the Einstein–Maxwell equations with small enough smooth and asymptotically flat initial data. We use harmonic gauge and Lorenz gauge. To cite this article: J. Loizelet, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Application d’Albanese pour les courbes et contractions

This article studies curves with singularities of “coordinate axes type”. As an application, we generalize a result of Deninger and Werner (Ann Sci école Norm Sup 38(4): 553–597, 2005), which is proved in the case of good reduction, to the case where the curve has semi-stable reduction.

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Résumé Cet article porte sur l’étude des courbes à singularité du type “axes de coordonnées” (TAC). Comme application, on généralise un énoncé de Deninger and Werner (Ann Sci école Norm Sup 38(4): 553–597, 2005), prouvé dans le cas de bonne réduction, au cas où la courbe a réduction semi-stable.