Showing papers in "Comptes Rendus De L Academie Des Sciences Serie I-mathematique in 1998"
TL;DR: In this article, the convergence du semi-groupe de Lax-Oleinik pour un lagrangien defini sur l'espace tangent d'une variete compacte, strictement convexe et super-lineaire dans les fibres.
Abstract: Resume Nous montrons la convergence du semi-groupe de Lax-Oleinik pour un lagrangien defini sur l'espace tangent d'une variete compacte, strictement convexe et super-lineaire dans les fibres.
235 citations
TL;DR: A new theory of “stochastic viscosity solutions” for fully nonlinear stochastic partial differential equations is proposed which allows to handle a large class of equations which covers in particular various applications such as models of phase transitions and front propagation in random media and pathwise Stochastic control.
Abstract: In this Note, we propose a new theory of “stochastic viscosity solutions” for fully nonlinear stochastic partial differential equations. This theory allows to handle a large class of equations which covers in particular various applications such as models of phase transitions and front propagation in random media and pathwise stochastic control. These applications will be detailed in a subsequent note.
183 citations
TL;DR: In this article, the authors present quelques applications of notre theorie au controle stochastique trajectoriel and a propagation de fronts dans des environnements aleatoires.
Abstract: Dans cette Note, nous etendons les resultats decrits dans une note precedente au cas d'Hamiltoniens non reguliers pour des equations aux derivees partielles stochastiques completement non lineaires. Ensuite nous presentons quelques applications de notre theorie au controle stochastique trajectoriel et a la propagation de fronts dans des environnements aleatoires.
160 citations
TL;DR: In this article, it was shown that for any p > 2 with p 2 N, the optimal version of (I p p ) is false if the scalar curvature of g is positive.
Abstract: Let ( M, g ) be a smooth compact Riemannian N -manifold, with N ≥ 2, and let p (1, N ) be real, and H 1 P ( M ) be the standard Sobolev space of order p . By the Sobolev embedding theorem, we have the inclusion H 1 p ( M ) ⊂ L p ⋆ )( M ), where p ⋆ = Np/(N - p ). Classically, this leads to some Sobolev inequality (I p 1 ), and then to some Sobolev inequality (I p p ), where each term in (I p 1 ) is elevated to the power p . Long standing questions were to know if the optimal versions with respect to the first constant of (I p 1 ) and (I p p ) do hold. Such questions received an affirmative answer by Hebey-Vaugon for p = 2. We prove here that, for p > 2 with p 2 N , the optimal version of (I p p ) is false if the scalar curvature of g is positive somewhere. In particular, there exist manifolds for which the optimal versions of (I p 1 ) is true, while the optimal version of (I p p ) is false. Among other results, we prove also that the assumption on the sign of the scalar curvature is minimal by showing that for any p (1, N ), the optimal version of (I p p ) holds on flat tori.
91 citations
TL;DR: The duality conjecture as mentioned in this paper states that the trajectories of a min-max function, considered as a dynamical system, have a linear growth rate (cycle time) and shows how this can be calculated through a representation of F as an infimum of max-plus linear functions.
Abstract: The set of min-max functions F : ℝn → ℝn is the least set containing coordinate substitutions and translations and closed under pointwise max, min, and function composition. The Duality Conjecture asserts that the trajectories of a min-max function, considered as a dynamical system, have a linear growth rate (cycle time) and shows how this can be calculated through a representation of F as an infimum of max-plus linear functions. We prove the conjecture using an analogue of Howard's policy improvement scheme, carried out in a lattice ordered group of germs of affine functions at infinity. The methods yield an efficient algorithm for computing cycle times.
85 citations
TL;DR: In this paper, the authors studied the long-time asymptotics of continuous-time branching random walk on Ωd (d ≥ 1) with a single source (i.e., branching site).
Abstract: We study the long-time asymptotics of continuous-time branching random walk on ℤd (d ≥ 1) with a single source (i.e., branching site). The random walk is assumed homogeneous, symmetric, irreducible, and having zero mean and finite variance of jumps. We find the limiting extinction probability and the asymptotics of all integer moments for the total population size and for the number of particles at a fixed site.
81 citations
TL;DR: In this article, the authors consider the problem of estimating the total energy of solutions in terms of the energy concentrated on the boundary, uniformly as the net-spacing h → 0.
Abstract: We consider the finite-difference and finite-element space discretization of the 1 — d wave equation with homogeneous Dirichlet boundary conditions in a bounded interval. We analyze the problem of estimating the total energy of solutions in terms of the energy concentrated on the boundary, uniformly as the net-spacing h → 0. We prove that there is no such a uniform bound due to spurious high frequencies. We prove however an uniform bound in suitable subspaces of solutions that eventually cover the whole energy space.
81 citations
TL;DR: In this paper, it was shown that I (λ, ·) has a bounded Palais-Smale sequence at level c (λ) for almost every λ, under hypotheses which do not ensure that the mountain-pass level c(λ) is a monotone function of λ.
Abstract: Let I (λ, ·), λ e ℝ, be a family of C 1 -functionals having mountain-pass geometry. Under hypotheses which do not ensure that the mountain-pass level c (λ) is a monotone function of λ, it is shown that I (λ) has a bounded Palais-Smale sequence at level c (λ), for almost every λ.
77 citations
TL;DR: In this paper, the authors deduisons des generalisations de theoremes de Bolotin ou Rabinowitz, and deduise le bon ensemble for trouver des connexions heteroclines.
Abstract: Resume Nous montrons que, pour un systeme dynamique lagrangien, le bon ensemble pour trouver des connexions heteroclines est l'ensemble de Peierls et non pas celui d'AubryMather. Nous en deduisons des generalisations de theoremes de Bolotin ou Rabinowitz.
75 citations
TL;DR: In this paper, the authors give a necessary and sufficient condition for the existence of control that keeps the corresponding trajectory of the related stochastic control system within a prescribed closed subset of the state space.
Abstract: In this Note, we give a necessary and sufficient condition for the existence of control that keeps the corresponding trajectory of the related stochastic control system within a prescribed closed subset of the state space. The problem of existence of stochastic control under a state-constraint is also called the viability property of the underlying control system. Our result is: the square of the distance function of this constraint is a viscosity supersolution of a Hamilton-Jacobi-Bellman equation if and only if the system enjoys the viability property.
69 citations
TL;DR: In this paper, robust invariant transitive sets containing singularities for C 1 flows on three-dimensional compact boundaryless manifolds are studied, where the transitive set cannot be destroyed by small C 1 -perturbations of the flow.
Abstract: The main goal of this paper is to study robust invariant transitive sets containing singularities for C 1 flows on three-dimensional compact boundaryless manifolds: they are partially hyperbolic with volume expanding central direction. Moreover, they are either attractors or repellers. Robust here means that this property cannot be destroyed by small C 1 -perturbations of the flow.
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TL;DR: In this article, a sufficient condition for recurrence of the d-dimensional stationary random walk defined by a Borel map is given, in terms of the asumptotic distributions of the maps (ƒ + ƒ T + … + fT n−1 /√ n converges in distribution to a Gaussian law on ℝ d ).
Abstract: Let T be a measure-preserving and ergodic automorphism of a probability space ( X, S, μ ). By modifying an argument in [4] we obtain a sufficient condition for recurrence of the d -dimensional stationary random walk defined by a Borel map ƒ : X → ℝ d ≥ 1, in terms of the asumptotic distributions of the maps (ƒ + ƒ T + … + ƒT n−1 )/ n 1/ d , n ≥ 1. If d = 2, and if ƒ: X → ℝ d satisfies the central limit theorem with respect to T (i.e. if the sequence (ƒ + ƒ T + … + fT n−1 /√ n converges in distribution to a Gaussian law on ℝ d ), then our condition implies that the two-dimensional random walk defined by ƒ is recurrent.
TL;DR: In this paper, the authors define a signe local Wp(E), defined by Deligne and Weil in terms of a certain nombre (fini) of courbes elliptiques (sur Q) auxiliaires.
Abstract: Resume Soit p un nombre premier A toute courbe elliptique E sur Qp correspond un signe local Wp(E), defini a partir des facteurs epsilon locaux de Deligne Ce signe est deja connu si p > 3 et aussi dans certains cas lorsque p = 2 ou 3 Dans cette Note, nous donnons la valeur de W2 ou W3 dans tous les cas Pour W3, nos resultats ne sont certains que si l'on admet qu'un certain nombre (fini) de courbes elliptiques (sur Q) auxiliaires sont des courbes de Weil
TL;DR: In this paper, the existence of a C γ e (0, ∞c) such that lim e → 0 e 2 / γ log ℡ ℙ ( sup 0 ≤ i ≤ 1 | B γ ( t ) | ≤ e ) is shown.
Abstract: Let B γ (t), 0 ≤ t ≤ 1 be a fractional Brownian motion of order γ e (0.2). and let B(t)=B 1 (t) be the standard Brownian motion. We show the existence of a C γ e (0, ∞c) such that: lim e → 0 e 2 / γ log ℙ ( sup 0 ≤ i ≤ 1 | B γ ( t ) | ≤ e ) = lim e → 0 e 2 / γ log ℙ ( sup 0 ≤ t ≤ 1 | W γ ( t ) | ≤ e / a γ ) = − C γ , . where a γ is an explicit constant and W γ ( t ) = 1 Γ ( ( γ + 1 ) / 2 ) ∫ 0 t ( t − s ) ( γ − 1 ) / 2 d B ( s ) . .
TL;DR: In this article, the behaviour of eigenvalues in problems which correspond to the vibrations of a drum, the whole mass of which is concentrated on a fractal subset of the drum, was studied.
Abstract: We study the behaviour of eigenvalues in problems which correspond to the vibrations of a drum, the whole mass of which is concentrated on a fractal subset of the drum.
TL;DR: In this paper, the authors consider existence, characterization, and calculation of equilibria in transportation networks, when the route capacities and demand requirements depend on time, and they express the problem in terms of a variational inequality and is situated in a Banach space setting.
Abstract: We consider existence, characterization, and calculation of equilibria in transportation networks, when the route capacities and demand requirements depend on time. The problem is expressed in terms of a variational inequality and is situated in a Banach space setting. © Academie des Sciences/Elsevier, Paris
TL;DR: In this paper, a preuve simple de phenomenes de regularite et de compacite induits par des noyaux de collision de Boltzmann singuliers is presented.
Abstract: Nous presentons une preuve simple de phenomenes de regularite et de compacite induits par des noyaux de collision de Boltzmann singuliers.
TL;DR: It is shown that with the approach discussed in this Note the authors can construct preconditioners spectrally equivalent to the original saddle-point matrix and leading to algorithms of optimal arithmetic complexity.
Abstract: In this Note we discuss the construction of efficient preconditioners for the solution of finite-dimensional saddle-point problems. A particular attention is given to those linear systems associated to the solution of elliptic problems by methods combining fictitious domain and distributed Lagrange multiplier techniques to force boundary conditions. It is shown that with the approach discussed in this Note we can construct preconditioners spectrally equivalent to the original saddle-point matrix and leading to algorithms of optimal arithmetic complexity.
TL;DR: In this article, the authors studied the Cauchy problem for the nonlinear wave equation with nonlinear damping aut|ut|m−1 and a nonlinear source term of type ba|a|p−1.
Abstract: We study the Cauchy problem tor the nonlinear wave equation with nonlinear damping aut|ut|m−1 and a nonlinear source term of type ba|a|p−1. When existence result is established for any initial data. When 1 < m < p. a finite time blow-up result is proved for compactly supported data with negative initial energy.
TL;DR: In this article, the Boltzmann equation is considered over a periodic spatial domain for bounded collision kernels, and appropriate scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L 1 ) to a unique limit governed by a solution of the acoustic or Stokes equations.
Abstract: The Boltzmann equation is considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L 1 ) to a unique limit governed by a solution of the acoustic or Stokes equations, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L 2 initial data of the acoustic or Stokes equations. The associated conservation laws are recovered in the limit.
TL;DR: In this paper, the authors confirm and explore a conjecture proposed by S. Peng regarding backward stochastic differential equations (in short BSDEs) and explore the conjecture in the context of this paper.
Abstract: This paper confirms and explores a conjecture proposed by S. Peng regarding backward stochastic differential equations (in short BSDEs).
TL;DR: In this article, l'existence de chocs faibles relatifs a valeur propre simple vraiment non lineaire du systeme and for un temps d'existence T independant de la force du choc.
Abstract: Resume Pour des systemes generaux de lois de conservation, en dimension quelconque d'espace, nous prouvons l'existence de chocs faibles relatifs a une valeur propre simple vraiment non lineaire du systeme et pour un temps d'existence T independant de la force du choc. En particulier, les resultats s'appliquent au systeme d'Euler de la dynamique des gaz.
TL;DR: In this paper, the first term of the asymptotic expansion of these spectral elements is solution of a spectral problem in a semifield of jets, which generalizes the max-algebra.
Abstract: We consider the asymptotics of the Perron eigenvalue and eigenvector of irreducible nonnegative matrices whose entries have a geometric dependance in a large parameter. The first term of the asymptotic expansion of these spectral elements is solution of a spectral problem in a semifield of jets, which generalizes the max-algebra. We state a “Perron-Frobenius theorem” in this semifield, which allows us to characterize the first term of this expansion in some non-singular cases. The general case involves an aggregation procedure a la Wentzell-Freidlin.
TL;DR: In this article, the resolution d'equations differentielles ordinaires for des champs de vecteurs peu reguliers a divergence nulle.
Abstract: Resume Nous etudions dans cette Note la resolution d'equations differentielles ordinaires pour des champs de vecteurs peu reguliers a divergence nulle. Apres avoir observe qu'il est equivalent de resoudre les equations de transport associees (i.e. les equations de Liouville), nous montrons des resultats d'existence, d'unicite et de stabilite pour des champs de vecteurs generiques dans L 1 ou pour des champs de vecteurs W 1.1 ≪ par morceaux ≫.
TL;DR: In this paper, le minima absolus de l'integrale d'action dans l'espace des lacets de moyenne nulle de classe H 1, sont exactement les solutions d'equilibre relatif dont la configuration est un minimum absolu du potentiel parmi les configurations dont le moment d'inertie par rapport a leur center de gravite est fixe.
Abstract: Resume Nous considerons le probleme des n corps dans un espace vectoriel euclidien E de dimension finie pour un potentiel d'attraction homogene invariant par isometrie, en particulier le potentiel newtonien. Nous montrons que les minima absolus de l'integrale d'action dans l'espace des lacets de moyenne nulle de classe H 1 dont la periode est fixee, sont exactement les solutions d'equilibre relatif dont la configuration est un minimum absolu du potentiel parmi les configurations dont le moment d'inertie par rapport a leur centre de gravite est fixe. Le resultat est a la fois general et elementaire et ne fait aucun usage de l'analyse fonctionnelle.
TL;DR: In this article, a caracterisation du spectre de Weyl au moyen des operateurs reguliers is described. Butteau et al. propose a new caracterization of the spectre of Weyl.
Abstract: Resume Dans cette Note, nous donnons une caracterisation du spectre de Weyl au moyen des operateurs reguliers. Ensuite, nous l’appliquons a l’equation de transport.
TL;DR: In this paper, the authors studied the range of the operator u ↦ ( | u ǫ | p − 2 u ′ ) ′ + λ 1 | u | p−2 u, u ( 0 ) = u ( T ) = 0, p > 1.
Abstract: In this work we study the range of the operator u ↦ ( | u ′ | p − 2 u ′ ) ′ + λ 1 | u | p − 2 u , u ( 0 ) = u ( T ) = 0 , p > 1. We prove that all functions h∈C1 [0,T] satisfying ∫ 0 T h ( t ) sin p π p t T d t = 0 lie in the range, but that if p ≢ 2 and h = 0, the solution set is bounded. Here sin p π p t T is a first eigenfunction associated to λ1.
TL;DR: In this paper, the densite des fonctions regulieres dans l'espace des champs de vecteurs L2 a divergence and rotationnel L2, and the trace tangentielle (ou normale) est L2 sur le bord Notre demonstration est basee sur des theoremes de regularite dans les domaines lipschitziens.
Abstract: Resume Nous montrons la densite des fonctions regulieres dans l'espace des champs de vecteurs L2 a divergence et rotationnel L2, et dont la trace tangentielle (ou normale) est L2 sur le bord Notre demonstration est basee sur des theoremes de regularite dans les domaines lipschitziens et constitue une simplification et une generalisation du resultat de [3]
TL;DR: In this paper, a huge classe de mesures ergodiques non necessairement homogenes, appelees g -mesures and for lesquelles des transitions de phases sont possibles.
Abstract: Resume Pour la dynamique symbolique sur un alphabet fini, la pression de Walters permet l'etude du formalisme multifractal d'une fonction continue. Les resultats obtenus ont un lien etroit avec la theorie des grandes deviations et s'appliquent a l'analyse multifractale classique d'une huge classe de mesures ergodiques non necessairement homogenes, appelees g -mesures et pour lesquelles des transitions de phases sont possibles.
TL;DR: The Dirichlet form associated with the intrinsic gradient on Poisson space is known to be quasi-regular on the complete metric space Γ = {ℤ+-valued Radon measures on ℝd}. as mentioned in this paper showed that under mild conditions, the set Γ\Γ is ǫ-exceptional, where Γ is the space of locally finite configurations in Ωd, that is, measures γ e Γ satisfying supxe, γ({x}) ≤ 1.
Abstract: The Dirichlet form associated with the intrinsic gradient on Poisson space is known to be quasi-regular on the complete metric space Γ = {ℤ+-valued Radon measures on ℝd}. We show that under mild conditions, the set Γ\Γ is ɛ-exceptional, where Γ is the space of locally finite configurations in ℝd, that is, measures γ e Γ satisfying supxe, γ({x}) ≤ 1. Thus, the associated diffusion lives on the smaller space Γ. This result also holds for Gibbs measures with superstable interactions.