Conference Stochastic Dynamics (SAMOS, 2007)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Ivan NOURDIN. Université Paris 6. Bande son disponible au format mp3 Durée : 4 mn

This talk does not suppose a priori that the evolution of a financial asset price is a semimartingale. The stochastic integral intervening in the definition of self-financing property is forward integral. If one requires that a certain minimal class of investor strategies are self-financing, previous prices are forced to be finite quadratic variation processes. The non-arbitrage property is not excluded if the class of admissible strategies is restricted. The classical notion of martingale is replaced with the notion of -martingale. Two instruments are developed: a calculus related to -martingales and infinite dimensional integration via regularization, with some examples. Some applications to no-arbitrage, viability, hedging and the maximization of the utility of an insider are expanded. The talk is essentially based on a joint work with Rosanna Coviello. Francesco RUSSO. Université Paris 13. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182789707066 (pdf) Bande son disponible au format mp3 Durée : 55 mn

Francesco RUSSO. Université Paris 13. Bande son disponible au format mp3 Durée : 4 mn

In this talk, we will describe some recent results concerning stochastic differential equations driven by a multidimensional fractional Brownian motion with Hurst parameter 1/3

Samy TINDEL. Université de Nancy. Bande son disponible au format mp3 Durée : 3 mn

In this work, we consider the Korteweg- de Vries equation perturbed by a random force of white noise type, additive or multiplicative. In a series of work, in collaboration with Y. Tsutsumi, we have studied existence and uniqueness in the additive case for very irregular noises. These use the functional framework introduced by J. Bourgain. We use similar tools to prove existence and uniqueness for a multiplicative noise. We are not able to consider irregular noises and have to assume that the driving Wiener process has paths in or . However, contrary to the additive case, we are able to treat spatially homogeneous noises. Then, we try to understand the effect of a small noise with amplitude on the propagation of a soliton. We prove that, on a time scale proportional to , a solution initially equal to the soliton but perturbed by a noise of the type above remains close to a soliton with modulated speed and position. The modulated speed and position are semi-martingales and we write the stochastic equations they satisfy. We prove also that a Central Limit Theorem holds so that, on the time scale described above, the solutions can formally be written as the sum of the modulated soliton and a gaussian remainder term of order . In the multiplicative case, we can go further. We prove that the gaussian part converges in distribution to a stationary process. Also, the equations for the modulation parameters allow to give a justification for the phenomenon called "soliton diffusion" observed in numerical simulations: the averaged soliton decays like . We obtain . This a joint work with A. de Bouard. Arnaud DEBUSSCHE. ENS Cachan. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182789894119 (pdf) Bande son disponible au format mp3 Durée : 45 mn

Arnaud DEBUSSCHE. ENS Cachan. Bande son disponible au format mp3 Durée : 2 mn

We study from a mathematical point of view a model equation for Bose Einstein condensation, in the case where the trapping potential varies randomly in time. The model is a nonlinear Schrödinger equation, with a quadratic potential with white noise fluctuations in time. We prove the existence of strong solutions in 1D and 2D in the energy space. The blow-up phenomenon will also be discussed under critical and super critical nonlinear interactions in the attractive case. This is a joint work with Reika Fukuizumi. Anne de BOUARD. Université Paris 11. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182789924489 (pdf) Bande son disponible au format mp3 Durée : 49 mn

We prove that an averaging principle holds for a general class of stochastic reaction-diffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite dimensional systems. Sandra CERRAI. Universita di Firenze. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182789980424 (pdf) Bande son disponible au format mp3 Durée : 47 mn

Sandra CERRAI. Universita di Firenze. Bande son disponible au format mp3 Durée : 6 mn

Some recently obtained results on (fBm)- driven linear and semilinear stochastic equations in infinite dimensional state spaces are reviewed. Regularity of the fractional Ornstein-Uhlenbeck process is studied and some results on large time behaviour are given (existence and ergodicity of stationary solutions, random fixed points) in the linear and semilinear case. The absolute continuity of measures induced by solutions is also studied. Bohdan MASLOWSKI. Academy of sciences of the Czech Republic. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182790040320 (pdf) Bande son disponible au format mp3 Durée : 45 mn

Bohdan MASLOWSKI. Academy of sciences of the Czech Republic. Bande son disponible au format mp3 Durée : 5 mn