Variance limite d'une marche aléatoire réversible en milieu aléatoire sur Z

The Central Limit Theorem for the random walk on a stationary random network of conductances has been studied by several authors. In one dimension, when conductances and resistances are integrable, and following a method of martingale introduced by S. Kozlov (1985), we can prove the Quenched Central Limit Theorem. In that case the variance of the limit law is not null. When resistances are not integrable, the Annealed Central Limit Theorem with null variance was established by Y. Derriennic and M. Lin (personal communication). The quenched version of this last theorem is proved here, by using a very simple method. The similar problem for the continuous diffusion is then considered. Finally our method allows us to prove an inequality for the quadratic mean of a diffusion. To cite this article: J. Depauw, J.-M. Derrien, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Moyennes limites et convexité

SoitF l'ensemble des fonctions bornées, définies surR ₊, à valeurs réelles non négatives, dont l'intégrale au sens de Lebesgue existe localement. A une fonctiona deF on associe le coupleL(a) des limites supérieure et inférieure de , quandt tend vers +. On pose

Position d'une marche aléatoire sur ℤ2 à l'instant d'atteinte d'une droite

Sans résumé     

Algébres topologiquement uniformes et algébres fortement topologiquement uniformes

We characterize locally convex topological algebrasA satisfying: a sequence (x _n) inA converges to 0 if, and only if, (x _n ²) converges to 0. We also show that a real Banach algebra such thatx _n ²+y _n ²→0 if, and only if,x _n → 0 andy _n → 0, for every sequences (x _n) and (y _n) inA, is isomorphic to, whereX is a compact space.      

Propriétés asymptotiques des processus de branchement en environnement aléatoire

Let (Z_n) be a branching process in a random environment. If the process in sub-critical or critical, we study the convergence rate of the survival probability P(Z_n>0) as n→∞; if the process is supercritical, we give a necessary and sufficient condition for the convergence in L^p of the natural martingale, where p>1 is given.     

Régularisation spectrale et propriétés métriques des moyennes mobiles

We develop a spectral regularization technique for moving averages , where ϕ is a nondecreasing map andU: H→H is a contraction of a Hilbert space (H, ‖·‖). We obtain a spectral regularization inequality which allows one to evaluate efficiently the increments ‖B _m ^U , ϕ (f)−B _n ^U , ϕ (f)‖,f∈H, by means of where is a properly regularized version of the spectral measure off with respect toU. We apply this inequality to an investigation of metric properties of the sets of moving averages {B _n ^{U, ϕ} (f), n ∈N} with fixedf∈H andN⊂ N. In particular, we obtain estimates of the associated covering numbers as well as of the related Littlewood-Paley-type square functions. This work extends our previous results concerning the case of classical averages (ϕ(n)=0). Since it is well-known that the structure of general moving averages is more complicated, there is no surprise that the general results we obtain are sometimes less complete than their classical counterparts and need suitable moment assumptions on the spectral measure (depending on the growth of the shift function ϕ). Nevertheless, when applied to the classical situation, our estimates still yield optimal bounds.

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Avec pour le premier author, le soutien de la fondation russe pour la recherche fondamentale, subvention 99-01-00112 et INTAS subvention 99-01317.     

Sur les fonctions convexes et les inégalités entre les valeurs moyennes

Sans résumé Conférence faite à la Société mathématique danoise le 17 janvier 1905.     

Cascades aléatoires et équations de Navier-Stokes

This work concerns the incompressible Navier-Stokes equation in. The non linear integral equation satisfied bx the Fourier transform of the Laplacian of the velocity field can be interpreted in terms of a branching Markov process and of a composition rule defined along the associated tree. We derive from this representation new classes where global existence and uniqueness can be proven.     

Marches aléatoires et diffusions dans les domaines lipschitziens

We give estimates for the transition kernel of a random walk and of a diffusion in a Lipschitz domain.     

Triangles inscrits et circonscrits à une courbe convexe sphérique

Sans résumé Présenté par G. Hajós