Monsieur Pierre-Yves Hénin, Président de l'Université Paris 1, acceuille des participants à la conférence et se félicite que le Centre Pierre Mendès-France serve de cadre à cette manifestation scientifique. Bande son disponible au format mp3 Durée : 6 mn

Conference Stochastic Dynamics (SAMOS, 2007)

01 - Intervention du président de l'Université Paris 1 - Pierre-Yves HENIN

Monsieur Cuong Le Van, Directeur du Centre d'Economie de la Sorbonne, présente ce centre en décrivant plus particulièrement les thématiques de recherche en mathématiques qui y sont développées. Bande son disponible au format mp3 Durée : 4 mn

02 - Présentation du Centre d'Economie de la Sorbonne - Cuong LE VAN

Consider the stochastic wave equation in dimension , , where denotes the formal derivative of a Gaussian stationary random field, white in time and correlated in space. Using Malliavin calculus, with Quer-Sardanyons we proved the existence and regularity of density of the law of the solution to the SPDE for any fixed . Denote this density by . More recently, with R. Dalang, we have established joint Hölder continuity in of the sample paths of the solution . On the basis of these two results, we can go further with the study of the properties in of the function , for any fixed . Using a method developed by Watanabe and applied to SPDEs in papers by Morien and Millet and Morien, we prove joint Hölder continuity of of the same order than the sample paths of the solution. We shall explain why the strong degeneracy of the fundamental solution leads to less regularity than one could have expected. Marta SANZ-SOLE. Universitat de Barcelona. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182789806745 (pdf) Bande son disponible au format mp3 Durée : 55 mn

03 - Property of the density for a three dimensional stochastic wave equation - Marta SANZ-SOLE

Marta SANZ-SOLE. Universitat de Barcelona. Bande son disponible au format mp3 Durée : 4 mn

04 - Questions - Marta SANZ-SOLE

The long term behaviour of dissipatively synchronized deterministic systems is determined by the system with the averaged vector field of the original uncoupled systems. This effect is preserved in the presence of environmental i.e., background or additive noise provided stochastic stationary solutions are used instead of steady state solutions. Random dynamical systems and random attractors provide the appropriate mathematical framework for such problems and require Ito stochastic differential equations to be transformed into pathwise random ordinary differential equations. An application to a system of semi-linear parabolic stochastic partial differential equations with additive space-time noise on the union of thin bounded tubular domains separated by a permeable membrane will be considered. What happens with linear multiplicative noise will also be considered. This a joint work with Tomas Caraballo (Sevilla) and Igor Chueshov (Kharkov). Based on the papers T. Caraballo and P.E. Kloeden, The persistence synchronization under environmental noise. Proc. Roy. Soc. London. A461 (2005), 2257-2267. T. Caraballo, I. Chueshov and P.E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain. SIAM J. Math. Anal. (to appear) Peter KLOEDEN. Johann Wolfgang Goethe University. Bande son disponible au format mp3 Durée : 39 mn

05 - Random Attractors and the Preservation of Synchronization in the Presence of Noise - Peter KLOEDEN

Peter KLOEDEN. Johann Wolfgang Goethe University. Bande son disponible au format mp3 Durée : 4 mn

06 - Questions - Peter KLOEDEN

We want to present some results on gradient systems with convex potential in finite and infinite dimension. The techniques are based on recent developments in the theory of gradient flows in the Wasserstein metric. (joint work with L. Ambrosio & G. Savaré). Lorenzo ZAMBOTTI. Université Paris 6. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182789954236 (pdf) Bande son disponible au format mp3 Durée : 43 mn

07 - Applications of optimal transport theory to stochastic equations - Lorenzo ZAMBOTTI

Lorenzo ZAMBOTTI. Université Paris 6. Bande son disponible au format mp3 Durée : 4 mn

08 - Questions - Lorenzo ZAMBOTTI

As a model for multiscale systems under random influences on physical boundary, a stochastic partial differential equation under a fast random dynamical boundary condition is investigated. An effective equation is derived and justified by reducing the random dynamical boundary condition to a usual random boundary condition. The effective system is still a stochastic partial differential equation, but is more tractable. Furthermore, the quantitative comparison between the solution of the original stochastic system and the effective solution is provided by estimating deviations. Jinqiao DUAN. Illinois Institute of Technology. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182790011791 (pdf) Bande son disponible au format mp3 Durée : 42 mn

09 - Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions - Jinqiao DUAN

Jinqiao DUAN. Illinois Institute of Technology. Bande son disponible au format mp3 Durée : 4 mn

10 - Questions - Jinqiao DUAN

We consider a directed polymer on the unit circle, with a continuous direction (time) parameter , defined as a simple random walk subjected via a Gibbs measure to a Hamiltonian whose increments in time have either long memory () or semi-long memory (), and which also depends on a space parameter (position/state of the polymer). is interpreted as the Hurst parameter of an infinite-dimensional fractional Brownian motion. The partition function of this polymer is linked to stochastic PDEs via a long-memory parabolic Anderson model. We present a summary of the new techniques which are required to prove that, in the semi-long memory case, converges to a positive finite non-random constant, and in the long-memory case, this limit is blows up, while the correct exponential growth function in that case is sandwiched between and . These tools include an almost sub-additivity concept, usage of Malliavin derivatives for concentration estimates, and an adaptation to the long-memory case of some arguments from the case (no memory), which require a detailed study of the interaction between the long memory, the spatial covariance, and the simple random walk. This talk describes joint work with Dr. Tao Zhang. Frederi VIENS. Purdue University. Bande son disponible au format mp3 Durée : 46 mn

11 - Almost-sure Lyapunov Exponent for a directed polymer in a fractional Brownian environment - Frederi VIENS

Frederi VIENS. Purdue University. Bande son disponible au format mp3 Durée : 4 mn

12 - Questions - Frederi VIENS

In this talk we consider the stochastic heat equation in , with vanishing initial conditions, driven by a Gaussian noise which is fractional in time, with Hurst index , and colored in space, with spatial covariance given by a function . Our main result gives the necessary and sufficient condition on for the existence of the process solution. When is the Riesz kernel of order this condition is , which is a relaxation of the condition encountered when the noise is white in space. When is the Bessel kernel or the heat kernel, the condition remains . This a joint work with C. Tudor. Raluca BALAN. University of Ottawa. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182790115349 (pdf) Bande son disponible au format mp3 Durée : 28 mn

13 - The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution - Raluca BALAN

Raluca BALAN. University of Ottawa. Bande son disponible au format mp3 Durée : 7 mn

14 - Questions - Raluca BALAN

In the first part of the talk, we will study the convergence of some weighted power variations of a fractional Brownian motion B. In the second part, we will apply the results obtained in the first part to compute the exact rate of convergence of some approximating schemes associated to scalar stochastic differential equations driven by B. In particular, we will be able to compute explicitly the limit of the error between the exact solution and the considered scheme. Ivan NOURDIN. Université Paris 6. Bande son disponible au format mp3 Durée : 41 mn

15 - Convergence of weighted power variations of fractional Brownian motion - Ivan NOURDIN

Enregistrement des conférences "Stochastic Dynamics" organisées par l'équipe de recherche SAMOS les 11 et 12 juin 2007. Les résumés des intervenants sont également disponibles sur l'Espace pédagogique interactif (http://epi.univ-paris1.fr/samos-conf-stochastic-dynamics). Recommandé à : étudiant de la discipline, chercheur - Catégorie : conférences - Année de réalisation : 2007

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Conference Stochastic Dynamics (SAMOS, 2007)

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