Modèles Stochastiques
en Temps Long
Titres et résumés
Michel Benaim, Neuchatel (Mini-cours)
Stochastic Algorithms.
Patrick Cattiaux, IMT, Toulouse (Mini-cours)
Some methods for the study of long time behaviour of stochastic dynamics. Application to McKean-Vlasov models.
In this mini-course we shall describe some of the usual methods for studying long time beahviour of diffusion type processes.
The first talk will be devoted to give an overview of various methods (functional inequalities, coupling and transportation inequalities) applied to the simplest possible diffusion processes: drifted brownian motion. The role of convexity will be particularly explained.
In the second one, we shall show how to extend these methods to more general diffusion processes including the case of interacting (linear) particle systems.
The third talk will focus on McKean-Vlasov type models, both from the stochastic and from the analytic points of view.
Gabriel Stoltz, CERMICS, Ecole des Ponts and MICMAC project team, INRIA Rocquencourt (Mini-cours)
Molecular dynamics: a mathematical introduction
I will present some standard approaches in computational statistical physics, and the associated mathematical results and issues. I will start by recalling the general framework of statistical physics (microscopic description of systems, thermodynamic ensembles and computation of average properties). I will then focus on the canonical ensemble, which can be sampled with Metropolis-Hastings algorithms or using realizations of the Langevin dynamics. Metastability however often prevents a direct computation of average properties, so that importance sampling strategies have to be resorted to in order to reduce the variance of the estimators under consideration. I will present one possible strategy, based on the use of the free energy associated with some slowly evolving degree of freedom. Finally, I will discuss the estimation of transport coefficients (such as the thermal conductivity, the shear viscosity, or the autodiffusion coefficient) by equilibrium or
nonequilibrium simulations.
Julien Barré, Laboratoire Dieudonné, Université de Nice
Cold atoms and non-linear Fokker-Plnack equations: modelin, theory and experiments. (Joint work with M. Chalony, B. Marcos and A. Olivetti from Nice and with D. Wilkowski from Nice and Singapur.)
In certain circumstances, a system of trapped cold atoms and quasi resonant lasers can be modeled as Brownian particles with effective long range interactions, and described by a non-linear Fokker-Planck equation. After briefly explaining the physical modeling, I will highlight two cases: (i) a quasi 1D trap where the effective interaction looks like 1D gravity, the theory can be compared to experiments done in Nice. (ii) a quasi 2D trap where the effective intercation does not derive from a potential and the non-linear Fokker-Planck equation may blow up in finite time.
François Bolley, Paris Dauphine (Exposé)
Long time behaviour for a McKean-Vlasov equation
We consider the issue of the long time behaviour of solutions to a McKean-Vlasov equation. When the exterior and interaction potentials are uniformly convex, then the solutions converge exponentially fast to a unique stationary solution, as shown by entropy dissipation and optimal transport techniques. Here we are interested in cases when the potentials are no more uniformly convex. This is joint work with I. Gentil (Lyon) and A. Guillin (Clermont-Ferrand).
Mathieu Faure, GREQAM, Marseille
Stochastic approximations and learning in Games (Exposé)
Stochastic approximations techniques, and in particular the ODE method, recently proved to be very useful in game theory, to predict the long run outcome in the situation of repeated interaction between players using adaptive strategies. After exposing the general framework, I will give some insights on two (more realistically one) recent result(s) in this context.
Benjamin Jourdain, CERMICS, ENPC (Exposé)
Optimal scaling of the transient phase of Metropolis Hastings algorithms (joint work with T. Lelièvre and B. Miasojedow)
We consider the Random Walk Metropolis algorithm on R^n with Gaussian
proposals, and when the target probability measure is the n-fold
product of a one dimensional law. It is well-known that, in the limit
n tends to infinity, starting at equilibrium and for an appropriate scaling
of the variance and of the timescale as a function of the dimension n,
a diffusive limit is obtained for each component of the Markov chain. We
generalize this result when the initial distribution is not the
target probability measure. The obtained diffusive limit is the solution
to a stochastic differential equation nonlinear in the sense of McKean.
We prove convergence to equilibrium for this equation. We discuss
practical counterparts in order to optimize the variance of the proposal
distribution to accelerate convergence to equilibrium. Our analysis
confirms the interest of the constant acceptance rate strategy (with
acceptance rate between 1/4 and 1/3).
Jérome Hénin, IBPC, CNRS/UPMC (Exposé)
Echelles de temps dans les simulations moléculaires en biologie :
problème et tentatives de solutions
La biologie est passée en quelques décennies de l'échelle macro et
mésoscopique (tissus, cellules), à une approche principalement
moléculaire, où nous cherchons à détailler les mécanismes du vivant
atome par atome. Ces détails échappent aux microscopes les plus
puissants, et ceux-ci sont de plus en plus remplacés par un
"microscope numérique" : la simulation de dynamique moléculaire. Aux
confins de la physique, de la chimie et de la biologie, les
simulations donnent une bonne description atomistique de
biomolécules comme les protéines, mais sont limitées à des échelles
de temps de l'ordre de la microseconde. A cette échelle, les
phénomènes intéressants correspondant à la fonction biologique des
protéines sont des événements rares. Nous illustrerons des approches
algorithmiques utiles pour observer ces événements rares dans le
cadre de problèmes biologiques.
Gilles Pagès, UPMC (Exposé)
Approximation of a stationary diffusion and related problems
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