Institut Élie Cartan de Lorraine

About: Institut Élie Cartan de Lorraine is a facility organization based out in Vandœuvre-lès-Nancy, France. It is known for research contribution in the topics: Boundary value problem & Stochastic differential equation. The organization has 345 authors who have published 1084 publications receiving 15512 citations. The organization is also known as: Institut Élie-Cartan de Nancy.

Integral estimation based on Markovian design

Abstract: Suppose that a mobile sensor describes a Markovian trajectory in the ambient space. At each time the sensor measures an attribute of interest, e.g., the temperature. Using only the location history of the sensor and the associated measurements, the aim is to estimate the average value of the attribute over the space. In contrast to classical probabilistic integration methods, e.g., Monte Carlo, the proposed approach does not require any knowledge on the distribution of the sensor trajectory. Probabilistic bounds on the convergence rates of the estimator are established. These rates are better than the traditional "root n"-rate, where n is the sample size, attached to other probabilistic integration methods. For finite sample sizes, the good behaviour of the procedure is demonstrated through simulations and an application to the evaluation of the average temperature of oceans is considered.

Contruction of quasi-invariant holomorphic parameters for the Gauss-Manin connection of a holomorphic map to a curve (second version)

Abstract: In this paper we consider holomorphic families of frescos (i.e. filtered differential equations with a regular singularity) and we construct a locally versal holomorphic family for every fixed Bernstein polynomial. We construct also several holomorphic parameters (a holomorphic parameter is a function defined on a set of isomorphism classes of frescos) which are quasi-invariant by changes of variable. This is motivated by the fact that a fresco is associated to a relative de Rham cohomology class on a one parameter degeneration of compact complex manifolds, up to a change of variable in the parameter. Then the value of a quasi-invariant holomorphic parameter on such data produces a holomorphic (quasi-)invariant of such a situation.

Geometrie tt* et applications pluriharmoniques

Abstract: Dans cette these nous introduisons la notion de fibre tt* (E,D,S), de fibre tt* metrique (E,D,S,g) et de fibre tt* symplectique (E,D,S,w) sur un fibre vectoriel E au-dessus d'une variete complexe, dans le langage de la geometrie differentielle reelle. Grâce a cette notion on obtient une correspondance entre des fibres tt* metriques et des applications pluriharmoniques admissibles de (M,J) dans l'espace symetrique pseudo-Riemannien GL(r,R)/O(p,q), avec (p,q) la signature de la metrique g. En utilisant ce resultat on obtient dans le cas ou M est compact Kahlerienne, un resultat de rigidite, puis un cas particulier du theoreme de Lu. De plus nous etudions des fibres tt* sur le fibre tangent TM et caracterisons une classe speciale qui contient les varietes speciales complexes et les varietes nearly Kahleriennes plates, et la sous-classe qui admet un fibre tt* metrique ou symplectique. En outre on analyse les fibres tt* qui proviennent de variations de structures de Hodge (VHS) et de fibres harmoniques. Pour les fibres harmoniques, la correspondance permet de generaliser un resultat de Simpson. L'application pluriharmonique associee a une variete specialement Kahlerienne reliee a l'application de Gaus duale, et celle associee a une VHS de poids impair est l'application de periodes. Si la structure complexe n'est pas integrable, on doit generaliser la notion de pluriharmonicite. Hors la rigidite ces resultats sont generalises au cas para-complexe.

A Probabilistic Look at Growth-Fragmentation Equations

Abstract: In this note, we consider general growth-fragmentation equations from a probabilistic point of view. Using Foster-Lyapunov techniques, we study the recurrence of the associated Markov process depending on the growth and fragmentation rates. We prove the existence and uniqueness of its stationary distribution, and we are able to derive precise bounds for its tails in the neighborhoods of both $0$ and $+\infty$. This study is systematically compared to the results obtained so far in the literature for this class of integro-differential equations.

On the qualitative behavior of solutions to certain stochastic partial differential equations of parabolic type

Abstract: This thesis is concerned with stochastic partial differential equations of parabolic type. In the first part we prove new results regarding the existence and the uniqueness of global and local variational solutions to a Neumann initial-boundary value problem for a class of non-autonomous stochastic parabolic partial differential equations. The equations we consider are defined on unbounded open domains in Euclidean space satisfying certain geometric conditions, and are driven by a multiplicative noise derived from an infinite-dimensional fractional Wiener process characterized by a sequence of Hurst parameters H = (Hi) i ∈ N+ ⊂ (1/2,1). These parameters are in fact subject to further constraints that are intimately tied up with the nature of the nonlinearity in the stochastic term of the equations, and with the choice of the functional spaces in which the problem at hand is well-posed. Our method of proof rests on compactness arguments in an essential way. The second part is devoted to the study of the blowup behavior of solutions to semilinear stochastic partial differential equations with Dirichlet boundary conditions driven by a class of differential operators including (not necessarily symmetric) Levy processes and diffusion processes, and perturbed by a mixture of Brownian and fractional Brownian motions. Our aim is to understand the influence of the stochastic part and that of the differential operator on the blowup behavior of the solutions. In particular we derive explicit expressions for an upper and a lower bound of the blowup time of the solution and provide a sufficient condition for the existence of global positive solutions. Furthermore, we give estimates of the probability of finite time blowup and for the tail probabilities of an upper bound for the blowup time of the solutions