Stochastic Problems in Mathematical Physics and Economics

Organisers: A. Agrachev (2016-2017), S. Kuksin, Y. Le Jan, V. Nersesyan, A. Shirikyan

Partially supported by ERC project DISPEQ and ANR project STOSYMAP

Organisers: A. Agrachev (2016-2017), S. Kuksin, Y. Le Jan, V. Nersesyan, A. Shirikyan

Partially supported by ERC project DISPEQ and ANR project STOSYMAP

26 Juin 2017, room 05, 3pm

Carlangelo Liverani (University Roma Tor Vergata, Italy): TBA

29 May 2017, room 05, 3pm

Raphael Lefevere (Université Paris Diderot, Paris, France): TBA

15 May 2017, room 421, 3pm

Freddy Bouchet (ENS de Lyon, France): TBA

24 April 2017, room 05, 3pm

Antonio Lerario (SISSA, Trieste, Italy): TBA

20 March 2017, room 421, 3pm

Massimiliano Gubinelli (Hausdorff Center for Mathematics, Bonn, Germany): Weak universality of fluctuations and singular stochastic PDEs

Abstract: Mesoscopic fluctuations of microscopic (discrete or continuous) dynamics can be described in terms of nonlinear stochastic partial differential equations which are universal: they depend on very few details of the microscopic model. Due to the extreme irregular nature of the random field sample paths, these equations turn out to not be well-posed in any classical analytic sense. In this talk I will review recent progress in the mathematical understanding of such singular equations and of their (weak) universality. If time permits I will discuss the case of the one dimensional Kardar-Parisi-Zhang equation and of the three dimensional Stochasic Allen-Cahn equation.

20 February 2017, room 421, 3pm

Alexandre Boritchev (Institut Camille Jordan, Université Lyon 1, France): Multidimensional Burgers Turbulence

Abstract: The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all models of turbulence. In particular, K41 and corrections to it provide estimates of small-scale quantities such as increments and energy spectrum for a 3D turbulent flow. However, because of the well-known difficulties involved in studying 3D turbulent flows, there are no rigorous results confirming or infirming those predictions.

Here, we consider a model for 3D turbulence: turbulence for the multi-dimensional potential Burgers equation. In the space-periodic case with a stochastic white in time and smooth in space forcing term, we give sharp estimates for small-scale quantities such as increments and energy spectrum and we obtain results on the speed of convergence to the stationary measure.

23 January 2017, room 05, 3pm

Darryl D. Holm (Imperial College London, Mathematics Department, UK): Variational Principles for Stochastic Fluid Dynamics

The approach is via a stochastic extension of the Hamiltons principle for fluid which imposes a constraint of stochastic transport of advected quantities, whose spatial correlations are obtained from observed data for tracer transport. The equations we derive via this approach keep their deterministic form and geometric meaning, which both derive from the variational principle. However, their transport vector field becomes stochastic, corresponding to stochastic Lagrangian particle paths. This means, for example, that Kelvin’s circulation theorem for the stochastically modified Euler equations for incompressible flow has the same integrand as in the deterministic case, but its circulation loop moves with the fluid flow along stochastic Lagrangian paths.

Details and examples for GFD may be found in: D.D. Holm, Variational principles for stochastic fluid dynamics, [2015] Proc Roy Soc A, 471: 20140963.

21 November 2016, room 421, 3pm

Filippo Santambrogio (Université Paris-Sud, France): Une introduction au transport optimal et aux flots de gradient dans l'espace de Wasserstein

Résumé: Le séminaire se composera essentiellement de trois parties. D'abord, pour introduire les équations d'évolution que je traiterai ensuite, je présenterai les généralités sur les flots de gradient dans l'espace Euclidien, sous la forme $x'(t)=-DF(x(t))$, et la manière d'adapter cette analyse à des espaces métriques.

Ensuite, je ferai une petite introduction au transport optimal, avec les problèmes de Monge, de Kantorovich, et la dualité, et aux distances de Wasserstein $W_p$ qu'on peut définir sur l'espace des mesures de probabilité en utilisant la valeur minimale d'un problème de transport.

Je terminerai en regardant quelles équations d'évolution sont en fait des flots de gradient dans l'espace des mesures de probabilité muni de la distance $W_2$, ce qui inclut l'équation de la chaleur, de Fokker-Planck, des milieux poreux..., ainsi que beaucoup d'autres EDP dans la dynamique des populations ou des gaz.

17 Octobre 2016, room 421, 3pm

Anne-Sophie de Suzzoni (Université Paris 13, Sorbonne Paris Cité, France): Invariant measures on the line for a class of Hamiltonian equations

Abstract: In this talk based on a joint work with F. Cacciafesta, I will consider a Hamiltonian equation whose Hamiltonian has a kinetic and a potential part. I will explain how to obtain a weakly invariant measure under the flow of this equation on the real line by applying cut-offs in space and in frequency and by localising the non linearity (or potential part of the Hamiltonian). These cut-offs and localisation reduce the problem to a finite dimensional one. The meaning to "weakly invariant" will be precised. The strategy uses two main ingredients : first, what one might refer to as the Prokhorov-Skorohod technique, which ensures that with uniform in the different cut-offs and localisation estimates on invariant measures on finite dimension, we get a weakly invariant measure on the line by passing to the limit; then, we use the Feynman-Kac theorem to get these estimates.

These ideas apply to a general range of equations. I will give an example with the Schrödinger equation with variable coefficients.

3 Octobre 2016, room 421, 3pm

Laurent Thomann (Université de Lorraine, France): Mesures invariantes pour les équations dispersives

Résumé: On commencera par faire des rappels sur la notion de mesure de probabilité invariante. On verra ensuite des applications à l'étude en temps grand de systèmes dynamiques. Dans un second temps, on présentera l'équation de niveau fondamental de Landau, et on construira des solutions globales à faible régularité à l'aide de mesures invariantes (Gibbs et bruit blanc).

5 September 2016, room 05, 3pm

Alexander Bufetov (Aix-Marseille Université & CNRS, France): Mesures conditionnelles des processus déterminantaux

Abstract: A cause de l'interaction non locale entre des particules, les propriétés dynamiques des mesures déterminantales sont très différentes de celles des mesures de Gibbs: par exemple, pour le sinus-processus, le nombre des particules dans un intervalle est presque surement détermine par la configuration en dehors de l'intervalle (rigidité de Ghosh-Peres).

Le résultat principal de cet expose est une formule explicite pour les mesures conditionnelles dans un intervalle par rapport a la configuration dans le complément de l'intervalle pour une classe assez large des processus déterminantaux en dimension 1. Le rôle clef est joue par la quasi-invariance de ces processus par le groupe de difféomorphismes au support compact. Pour le cas spécial du processus au noyau Gamma, dans le cadre discret, la quasi-invariance par le groupe symétrique infini a été établi par Olshanski; le cas général sera traité dans l'expose qui se base sur deux prépublications.

[1] A.I. Bufetov, Quasi-Symmetries of Determinantal Point Processes.

[2] A.I. Bufetov, Conditional measures of determinantal point processes.

20 June 2016, room 05, 3pm

Thierry Bodineau (École Polytechnique, France): Large deviations and non-equilibrium statistical mechanics

Abstract: In this talk, we will review some results on the steady states of diffusive systems maintained off equilibrium by two heat baths at unequal temperatures. Using the framework of the hydrodynamic limits, we will discuss the large deviations of the heat current through these systems. In particular, we will explain the occurrence of dynamical phase transitions which may occur for some models.

13 June 2016, room 05, 3pm

Noé Cuneo (University of Genève, Switzerland): Non-equilibrium steady states for chains of rotors

Abstract: We study the existence of an invariant measure (called non-equilibrium steady state) for chains of rotors interacting with stochastic heat baths at different temperatures. The interesting issue is that rotors with high energy tend to decouple from their neighbors, due to the fast oscillations of the interaction forces. This phenomenon makes the existence of a steady state challenging to prove, and prevents the system from relaxing exponentially rapidly to its invariant measure. I will introduce the model and recall some results about Lyapunov functions. Then, I will sketch the proofs that we have written with J.-P. Eckmann and C. Poquet for chains of length 3 and 4. Finally, I will explain why we cannot handle longer chains (for now!).

9 May 2016, room 05, 3pm

Konstantin Khanin (University of Toronto, Canada): On global solutions to the random Hamilton-Jacobi equation

Abstract: We shall discuss a problem of existence and uniqueness of global solutions to the random Hamilton-Jacobi equation. While the problem in the spatially periodic setting is well understood by now, the situation in the non-compact case remains largely open.

In this talk we shall discuss both cases, and present partial results and conjectures in the non-compact situation. We shall also discuss a connection with the problem of KPZ universality.

4 April 2016, Amphi Darboux, 3pm

Sergio Simonella (Technische Universität München, Germany): Evolution of marginals in the hard sphere system

Abstract: The problem of the rigorous validity for the Boltzmann kinetic equation knows no other approach than a detailed study of the BBGKY hierarchy, namely the complete set of equations for the evolution of marginals in a large system of interacting particles. Focusing on the paradigmatic case of hard spheres, I will describe the hierarchy and the structure of its solution. Then I will discuss some method for its derivation within minimal regularity assumptions.

14 March 2016, room 05, 3pm

Nicolai Krylov (University of Minnesota, USA): On the existence of ${\bf W}_p^2$ solutions for fully nonlinear elliptic equations under relaxed convexity assumptions

Abstract: We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains without requiring $H$ to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of $H$ at points at which $|D^{2}v|\leq K$, where $K$ is any given constant. For large $|D^{2}v|$ some kind of relaxed convexity assumption with respect to $D^{2}v$ mixed with a VMO condition with respect to $x$ are still imposed. The solutions are sought in Sobolev classes.

16 February 2016, room 314, 4pm

Laszlo Erdös (Institute of Science and Technology, Austria): Universality of spectral statistics of random matrices, Part II

15 February 2016, room 314, 3pm

Laszlo Erdös (Institute of Science and Technology, Austria): Universality of spectral statistics of random matrices, Part I

Abstract: E. Wigner predicted that the local eigenvalue gap statistics of sufficiently complex quantum systems are universal. One prominent model to test this theory is matrices with independent random entries. In the recent years the universality conjecture for a very broad class of random matrices has been resolved. In these talks I will present the main questions and explain the novel analytical techniques that have been developed to solve them.

1 February 2016, room 05, 3pm

Mark Freidlin (University of Maryland, USA): Large-time effects of small perturbations and the simplex of invariant measures

Abstract: Perturbations of dynamical systems and semi-flows will be considered. Long-time evolution of the perturbed dynamics, under some assumptions, consists of a slow component, which is, actually, a motion on the simplex of invariant probability measures of the non-perturbed system, and of a fast component which can be characterized by the invariant measure corresponding to the slow component position. The slow component, in an appropriate time scale, converges weakly to a motion on the simplex. This limiting slow motion can be stochastic even in the case of pure deterministic perturbations of deterministic systems. I will consider various realizations of this general approach for perturbations of systems defined by ODEs and PDEs.

18 January 2016, room 05, 3pm

Shizan Fang (Université de Bourgogne, France): Navier-Stokes equations on Riemannian manifolds

Abstract: On a Riemannian manifold, there exist several "Laplacian operators" on vector fields. In this talk, we will present different probabilistic behaviors behind these equations.

11 January 2016, room 05, 3pm

Isabelle Gallagher (Université Paris-Diderot, France): De la dynamique moléculaire aux équations de l’acoustique et de Stokes-Fourier

Résumé: La question du passage d'une description microscopique de particules (via la mécanique déterministe newtonienne) à une description macroscopique (via des équations de la mécanique des fluides) est un problème largement ouvert. Dans cet exposé nous présenterons quelques progrès récents dans des régimes linéaires, obtenus avec Thierry Bodineau et Laure Saint-Raymond.

14 December 2015, room 201, 3pm

Giambattista Giacomin (Université Paris Diderot, France): Weak noise and non hyperbolic unstable fixed points

Abstract: We consider one dimensional ordinary stochastic differential equations driven by additive Brownian motion with small variance. When the variance is zero such equations have an unstable non-hyperbolic fixed point and the drift near such a point has a power law behavior. For positive variance, the fixed point property disappears, but it is replaced by a random escape or transit time which diverges as the variance tends to zero. We show that this random time, under suitable (easily guessed) rescaling, converges to a limit random variable that (essentially) depends only on the power exponent associated to the fixed point. Such random variables, or laws, have therefore a universal character and they arise of course in a variety of contexts. We obtain quantitative sharp estimates, notably tail properties, on these universal laws. Work in collaboration with M. Merle.

16 Novembre 2015, room 201, 3pm

Juraj Földes (Université Libre de Bruxelles, Belgium): Large parameter limits for stochastically forced equations

Abstract: Due to sensitivity with respect to initial data and parameters, individual solutions of the basic equations of fluid mechanics are unpredictable and seemingly chaotic. However, some of their statistical properties are robust. As early as the 19th century it was conjectured that turbulent flow cannot be solely described by deterministic methods, and indicated that a stochastic framework should be used. In this framework, invariant measures of the stochastic equations of fluid dynamics presumably contain the statistics posited by the basic theories of turbulence.

In this talk we investigate properties of invariant measures for the Boussinesq equations and Magneto-hydrodynamic equations in the presence of a degenerate stochastic forcing acting only in the temperature component. The main goal is to prove convergence of invariant measures in singular limits when Prandtl or Rossby and magnetic Reynolds numbers approach infinity.

More precisely, we show a general framework for converting the problem of convergence of measures to the question of finite time convergence of solutions. Then we analyze singular limit problems in a stochastic setting. This is a joint work with S. Friedlander (U. of Southern California), N. Glatt-Holtz (Virginia Tech), G. Richards (U. of Rochester), and E. Thomann (Oregon State).

19 October 2015, room 05, 3pm

Jani Lukkarinen (University of Helsinki, Finland): Hydrodynamics without scaling limits: Thermalization in harmonic particle chains with velocity flips

Abstract: Hydrodynamics, and the closely related Fourier's law of heat conduction, are successful models for macroscopic transport in many physical systems. Although complete understanding of their microscopic origin is still missing, solid progress has been made in deriving macroscopic and mesoscopic evolution equations using scaling limits. However, the scaling limits sometimes offer only a partial picture of the dynamics behind the transport phenomena. One issue obscured by hydrodynamic scaling limits is "thermalization". This refers to the evolution of an initial state, assumed to be sufficiently chaotic but otherwise fairly unrestricted, into a local equilibrium state which determines the initial data of the hydrodynamic evolution equations. The thermalization process is typically hard to control rigorously as it involves evolution at the microscopic scales, while the hydrodynamic evolution occurs at time scales $O(L^2)$, where $L$ denotes the spatial length scale of the system.

In this talk, I will discuss thermalization in a particle chain where the particles interact via a harmonic potential and each particle flips the direction of its velocity randomly. It turns out that after a relatively short time, the average kinetic temperature profile satisfies the Fourier's law, in a local microscopic sense, without assuming that the initial data is close to a local equilibrium state. The bounds derived here prove that the thermalization period is at most of the order of $L^{2/3}$ where L denotes the number of particles in the chain. In particular, even before the diffusive time scale, Fourier's law becomes a valid approximation of the evolution of the kinetic temperature profile. The talk is mainly based on J. Stat. Phys. 155 (2014) 1143-1177 (arxiv.org/abs/1308.4901).

5 October 2015, room 05, 3pm

Vahagn Nersesyan (Université de Versailles, France): The Gallavotti-Cohen principle for randomly forced PDE's

Abstract: The mathematical, physical, and numerical aspects of the Gallavotti-Cohen principle are widely studied in the literature. The previous rigorous mathematical works are mainly concerned with finite-dimensional stochastic systems. The aim of this talk is to present a result for an infinite-dimensional model provided by the randomly forced Burgers equation. Assuming that the force is rough with respect to the space variables and has a non-degenerate law, we prove a large deviations principle (LDP) for some unbounded functionals. Then in the case of the entropy production functional, we show that the rate function of the LDP satisfies a symmetry of Gallavotti-Cohen type. This is a joint work with V. Jaksic, C.-A. Pillet and A. Shirikyan.

15 June 2015, room 421, 3pm

Vojkan Jaksic (McGill University, Canada): Physics and mathematics of fluctuation relations

Abstract: The discovery of fluctuation relations revolutionized our understanding of non-equilibrium statistical mechanics. In this talk, I will try to describe their physical and mathematical significance in simplest possible terms.

8 June 2015, room 421, 3pm

Stanislav Molchanov (University of North Carolina at Charlotte, USA): Anderson parabolic problem with the random non-stationary potentials, localization versus intermittency (review)

Abstract: a) Stationary potentials, relations with the Anderson localization in the solid state physics. Quenched and annealed asymptotics for the moments (by examples).

b) Non-stationary potentials ergodic in space and short correlated in time. Magnetic field in such (turbulent) velocity fields (dynamic problem). Intermittency.

c) Population dynamics. KPP model with propagating front. Intermittency inside the front. Instability of the steady states with respect to the small random perturbations.

d) Open problems.

18 May 2015, room 421, 3pm

Olivier Glass (Université Paris-Dauphine): Contrôlabilité pour l'équation d'Euler non isentropique

Résumé: On considère la question de contrôlabilité de l'équation d'Euler non isentropique pour les gaz compressibles polytropiques, dans le contexte de solutions faibles d'entropie à petite variation totale. On considère le système à la fois en variables lagrangiennes et en variables euleriennes, et on obtient un résultat de contrôlabilité frontière dans les deux cas.

13 April 2015, room 421, 3pm

Vahagn Nersesyan (Université de Versailles): Local large deviations principle for occupation measures of the damped nonlinear wave equation perturbed by a white noise

Abstract: In this talk, we will consider the damped nonlinear wave (NLW) equation driven by a spatially regular white noise. Assuming that the noise is non-degenerate in all Fourier modes, we will establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation. The proof relies on a generalization of methods developed in [JNPS1] and [JNPS2] in the context of kick forced dissipative PDE’s with parabolic regularization property such as, for example, the Navier–Stokes system or the complex Ginzburg–Landau equations. We show that a high concentration towards the stationary measure is impossible, by proving that the rate function that governs the LDP cannot have the trivial form (i.e., vanish on the stationary measure and be infinite elsewhere). This is a joint work with D. Martirosyan.

[JNPS1] V. Jaksic, V. Nersesyan, C.-A. Pillet, A. Shirikyan, Large deviations from a stationary measure for a class of dissipative PDE’s with random kicks, Comm. Pure Appl. Math. 68 (2015).

[JNPS2] V. Jaksic, V. Nersesyan, C.-A. Pillet, A. Shirikyan, Large deviations and mixing for dissipative PDE’s with unbounded random kicks, preprint, 2014.

23 March 2015, room 421, 3pm

Elena Kartashova (Johannes Kepler University, Linz, Austria): Wave turbulence theory: historical overview and open questions

Abstract: The beginning of the wave turbulence theory (WTT) is considered to be 1967, when it was first discovered that the so-called wave kinetic equation has stationary solutions. Stationary solutions, given by power law functions, describes the energy spectrum of a weakly nonlinear wave system in the Fourier space. The search for such spectra and the study of their properties is the subject of kinetic WTT. The next important achievement in this area relates to 1990, when it was discovered the existence of independent resonant clusters. The problem of constructing of the resonance cluster set for given dispersive wave system can be reduced to solving of a Diophantine equation in several variables in high powers. The construction of resonance clusters and the study of their properties is the subject of discrete WTT.

16 March 2015, Lecture theater Hermite, 4pm

Herbert Spohn (Zentrum Mathematik, TU München): The noisy Burgers equation with several components

Abstract: My interest is the noisy Burgers equation on the entire real line, the noise being space-time white noise. In particular I would like to understand the structure of the stationary covariance. I recall the exact solution in the scalar case. The case of several components is of great physical interest with very few mathematical results only. Conjectures and preliminary results will be discussed.

9 March 2015, room 421, 3pm

Jürg Fröhlich (ETH - Institute for Theoretical Physics): Effective dynamics in quantum theory

Abstract: I will discuss some important examples of effective (stochastic) dynamics in quantum theory:

(1) A model of a quantum particle coupled to a heat bath at positive temperature; the theorem being that the particle exhibits Quantum Brownian Motion (work with De Roeck and Pizzo);

(2) A simplified treatment of (1); but with a random external potential added (Anderson model coupled to a heat bath); the theorem being that thermal noise destroys Anderson localization (work with Schenker);

(3) A mesoscopic quantum system subjected to repeated projective measurements of a single "observable"; our results concerning a theory of "events and quantum jump processes" (work with Ballesteros, Fraas and Schubnel).

9 February 2015, room 421, 3pm

Andrey Dymov (Université de Cergy-Pontoise): Nonequilibrium statistical mechanics of solids in medium

Abstract: Investigation of the energy transport in solids is one of the main problems in the nonequilibrium statistical mechanics. Since it turns out to be extremely difficult, usually one studies toy models, possessing additional ergodic properties. A common idea is to consider a Hamiltonian system of particles where each mode is a subject to stochastic perturbation. Clearly, it is important to study the case when the perturbation goes to zero.

In this talk I will discuss dynamics of an anharmonic system of weakly interacting oscillators, where each oscillator is weakly coupled with its own stochastic Langevin thermostat. The system can be interpreted as a solid plugged in medium and weakly interacting with it. I will prove that, under the limit when the couplings of oscillators with each other and with the thermostats go to zero with some precise scaling, behaviour of the system is governed by an effective equation which is a rather nice dissipative SDE. I will show that under the limit above, dynamics of the energy satisfies laws, which resemble the Fourier law and the Green-Kubo formula (but which are not the F. law and the G.-K. formula).

1 December 2014, room 421, 3pm

François Golse (Ecole polytechnique, Paris): Mean field limits and kinetic models

Abstract: Various kinetic models, such as the Vlasov-Poisson system, are formally derived from the dynamics of a large number of interacting particles in some scaling limit known as the mean field limit. There are two different approaches to this type of limit: one involves the notion of empirical measure of the particle system in the single-particle phase space, while the other is based on a procedure known as the BBGKY hierarchy.

The first part of the talk (work in collaboration with C. Mouhot and V. Ricci) explains how both approaches are related, and discusses convergence rates based on the work of R.L. Dobrushin (Func. Anal. 1979). The second part of the talk explains how Dobrushin's estimate can be extended to a variant of the Vlasov-Maxwell system.

17 November 2014, room 421, 3pm

Marcello Porta (University of Zürich): Mean-field evolution of fermionic systems

Abstract: In this talk I will discuss the dynamics of interacting fermionic systems in the mean-field regime. Compared to the bosonic case, fermionic mean-field scaling is naturally coupled with a semiclassical scaling, making the analysis more involved. From a physical point of view, as the number of particles grows one expects the quantum evolution of the system to be effectively described by Hartree-Fock theory. The next degree of approximation is provided by a classical effective dynamics, corresponding to the Vlasov equation.

I will consider initial data which are close to quasi-free states, both at zero and at positive temperature, with an appropriate semiclassical structure. Under some regularity assumptions on the interaction potential I will show that the time evolution of such initial data stays close to a quasi-free state, with reduced one-particle density matrix given by the solution of the time-dependent Hartree-Fock equation. The result holds for all (semiclassical) times, and gives effective bounds on the rate of convergence towards the Hartree-Fock dynamics as the number of particles goes to infinity.

20 October 2014, room 421, 3pm

Viviane Baladi (ENS, Paris): The spectrum of Sinai billiard flows

Abstract: Sinai billiard maps in dimension two have been known to be exponentially mixing (L.-S. Young) for almost two decades, and recent work of Demers and Zhang have shed new light on the spectrum of their transfer operators. The situation for the continuous time Sinai billiard is more delicate. I will present recent results and ongoing work on their spectrum. (Joint work with M. Demers and C. Liverani).

6 October 2014, room 421, 3pm

Frédéric Rousset (Université de Paris-Sud, Orsay): L’amortissement Landau pour un modèle simple de particules en interaction

Résumé : Le but de l’exposé sera de présenter, pour le modèle très simple de Vlasov-HMF qui décrit des particules sur le cercle interagissant avec un potentiel régulier, des résultats d’amortissement Landau dans des espaces de Sobolev. Il s’agit de décrire en temps grand pour une équation hamiltonienne et réversible le comportement des solutions qui sont des petites perturbations d’équilibres spatialement homogènes stables. Travail en commun avec Erwan Faou.

12 May 2014, room 421, 3pm

Sergey Nazarenko (University of Warwick): Open questions in Wave Turbulence theory

Abstract: I will outline the main steps in the Wave turbulence approach and highlight some open questions for mathematicians.

7 April 2014, room 421, 3pm

Wei-Min Wang (Université de Cergy-Pontoise): Witten Laplacian and Stochastic NLS

Abstract: We shall discuss some recent works on stochastic NLS on the torus, which proves exponential approach to equilibrium. The most precise results are obtained on the circle, both for the focusing and the defocusing cases. (These are joint works with Carlen, Froehlich and Lebowitz in various combinations.)

24 March 2014, Amphithéâtre Hermite, 3pm

Nicolas Burq (Université de Paris-Sud, Orsay): Gibbs measures and NLS on planar domains

Abstract: In this talk I will present the construction of Gibbs measures and Wick reordered NLS on bounded planar domains (or more generally any compact surface with boundary). I will also show how these measures allow to construct a weak flow for NLS, and show some perspectives toward strong flows. This is a joint work with L. Thomann and N. Tzvetkov.

10 February 2014, room 421, 3pm

Thierry Bodineau (École Polytechnique, Palaiseau): Diffusion pour une particule marquée dans un gaz dilué de sphères dures

Résumé : On étudie le mouvement d’une particule marquée dans un gaz dilué de sphères dures à l’équilibre. Après changement d’échelle de l’espace et du temps, on montre que cette particule suit un mouvement brownien. (travail commun avec I. Gallagher et L. Saint-Raymond)

20 January 2014, room 05, 3pm

Yves Le Jan (Université de Paris-Sud, Orsay): Lacets markoviens

Résumé : On présentera quelques résultats anciens et nouveaux sur les ensembles Poissoniens de lacets markoviens, leurs amas, et leurs relations avec le champ libre.

16 December 2013, room 421, 3pm

Denis Bernard (LPT-ENS, Paris): On the relation between SLE and CFT (or, more generally, statistical mechanics)

Abstract: Stochastic Schramm Loewner Evolutions (SLE) are Markov processes describing fractal curves or interfaces in two-dimensional critical systems. After a presentation of the objects SLE deals with, and of the basics tools involved in their description, I will present the essential points underlying the connection between statistical mechanics and processes which, in the present context, leads to a connection between SLE and conformal field theory (CFT). The lecture is intended to be at an introductory level.

2 December 2013, room 421, 3pm

Alexandre Boritchev (Université de Genève): Hyperbolicité des minimiseurs pour l'équation de Burgers stochastique

Résumé : Nous regardons l'équation de Burgers stochastique du point de vue lagrangien. En d'autres mots, nous étudions le comportement dynamique des minimiseurs d'énergie qui induisent la description variationnelle des solutions. Sous des conditions de non-dégénérescence sur le forçage aléatoire, nous prouvons l'hyperbolicité de ces minimiseurs. Nous simplifions considérablement la preuve donnée dans l'article [E, Khanin, Mazel, Sinai, Annals, 2000]. Nous conclurons par le lien entre ce problème et celui de la convergence vers la mesure stationnaire pour les solutions de l'équation.

Travail en collaboration avec K. Khanin (Université de Toronto)

18 November 2013, room 421, 3pm

Armen Shirikyan (University of Cergy-Pontoise): Large deviations from a stationary measure for dissipative PDE's with random kicks

Abstract: We study a class of dissipative PDE's perturbed by a random kick force. It is well known that if the random perturbation is sufficiently non-degenerate, then the Markov process associated with the problem in question has a unique stationary distribution, which is exponentially mixing. In addition, the strong law of large numbers and the central limit theorem hold and give the large-time behaviour of probabilities for small deviations of the time average of continuous functionals from their spatial average with respect to the stationary distribution. In this talk, I discuss the asymptotics of probabilities of order-one deviations from the stationary measure. Our main result shows that the occupation measures of solutions satisfy the LDP with a good rate function. The proof is based on Kifer's criterium for LDP, a Lyapunov-Schmidt type reduction, and a general result on long-time behaviour of generalised Markov semigroups. The result applies to the 2D Navier-Stokes system, Ginzburg-Landau equation, and other dissipative PDE's. This is a joint work with V. Jaksic, V. Nersesyan, and C.-A. Pillet.

4 November 2013, room 421, 3pm

Massimiliano Gubinelli (University Paris-Dauphine): SPDEs, paraproducts and all that

Abstract: In this talk we explain recent advances in understanding of the functional analytic structure of solutions to non-linear SPDEs and their application to the study of various model of mathematical physics: a 2d parabolic Anderson model, the 3d stochastic quantisation equation and the 1d Kardar-Parisi-Zhang equation. These advances have been possible thanks to a generalisation of the theory of controlled rough paths. In particular we discuss the role of multiscale decomposition of distributions and of the notion of paraproduct in the analysis of this problem. I will assume only basic knowledge of functional analysis, in particular no results of rough path theory or stochastic analysis are needed to follow the talk.

14 October 2013, room 01, 3pm

Tomasz Komorowski (Polish Academy, Warsaw): Long time energy transfer in the random Schrodinger equation

Abstract: We consider the long time behavior of solutions of the d-dimensional linear Boltzmann equation that arises in the weak coupling limit for the Schrodinger equation with a time-dependent random potential. We show that the intermediate mesoscopic time limit satisfies a Fokker-Planck type equation with the wave vector performing a Brownian motion on the (d − 1)-dimensional sphere of constant energy, as in the case of a time-independent Schrodinger equation. However, the long time limit of the solution with an isotropic initial data satisfies an equation corresponding to the energy being the square root of a Bessel process of dimension d/2. This is a joint work with Lenya Ryzhik.

30 September 2013, room 05, 3pm

Stefano Olla (University Paris-Dauphine): From dynamics to thermodynamics

Abstract: One of the purposes of statistical mechanics is to explain thermodynamics in terms of ‘microscopic’ dynamics. The general consensus is that thermodynamics describe the ‘macroscopic’ behaviour of some quantities (energy, pressure, …) under the influence of external forces or thermostats. Macroscopic means that laws of thermodynamics emerge for ‘large’ systems and in a large time scale.

I will present a precise mathematical approach where thermodynamics laws (Carnot cycles, Kelvin and Clausius principles, etc.) are obtained through space-time scaling limits. In this approach, that I consider ultraorthodox, the problem of thermodynamics, and in particular its second principle, is reduced to a purely mathematical problem, very difficult indeed.

Avoiding a general theory, I will talk about the simplest system: a ‘one dimensional’ rubber subject to tension and heat bath. This will be modelled microscopically by a (classical) chain of oscillators (of Fermi-Pasta-Ulam type) subject to boundary forces and eventually stochastic Langevin thermostats.

Irreversible isothermal and adiabatic transformations can be obtained for the evolution of ‘local’ energy and stress, by hydrodynamic limits, performing space-time scalings. Quasi-static reversible thermodynamic transformations, object of the classical thermodynamics, are then obtained by further time rescaling.

From the mathematical point of view, isothermal transformation can be fully proven, while adiabatic transformations are the main challenge.

16 September 2013, room 01, 3pm

Alberto Maiocchi (University of Cergy-Pontoise): Adiabatic invariants in the thermodynamic limit

Abstract: I will show how to construct an adiabatic invariant for a large 1-d lattice of particles, such that the time evolution of this quantity is negligible during exponentially long times as the temperature goes to zero. In difference with the results available in the literature, the present result holds uniformly in the thermodynamic limit, when the number of degrees of freedom tends to infinity, while the temperature stays fixed.

In the first part of the talk I will present an introduction to the problem and state the main result, stressing the consequences for physics and giving a sketch of the proof. The second part will be devoted to the illustration of the techniques used to control the decay of the spatial correlations with respect to the Gibbs measure, as they play a key role in the proof. This techniques is an adaptation of the Dobrushin method.

1 July 2013, room 01, 3pm

Arnaud Debussche (ENS Cachan - Bretagne): Comportement en temps long pour les lois de conservation stochastiques

Résumé : Il s'agit d'un travail en commun avec J. Vovelle dans lequel nous étudions le comportement en temps long des solutions de lois de conservations stochastiques. L'existence de celles-ci a été obtenue par plusieurs auteurs. Nous avons obtenue un résultat très général grace à une généralisation au cadre stochastique de la formulation cinétique introduite par Lions, Perthame et Tadmor. Cette formulation est très puissante car elle permet de garder trace de la dissipation. En utilisant un lemme de moyenne, nous parvenons à montrer que, si l'equation n'est pas dégénérée, la dissipation d'énergie est suffisante pour assurer l'existence d'une mesure invariante. De plus, en dimension un et pour des flux au plus quadratique nous montrons qu'il y a unicité de la mesure invariante, et donc ergodicité. Nous généralisons ainsi un résultat de E, Khanin, Mazel et Sinai, obtenu pour l'équation de Burgers, à des équations générales pour lesquelles la formule de Hopf-Lax-Oleinik n'est pas valide.

17 June 2013, room 05, 3pm

Vojkan Jaksic (University of McGill): Non-equilibrium statistical mechanics of the spin-boson model

Abstract: The non-equilibrium spin-boson model describes interaction of a quantum dot (say, spin 1/2) with several independent bosonic thermal reservoirs which are initially in thermal equilibrium at distinct temperatures. The temperature differentials result in non-trivial energy/entropy flux across the system. In this talk we will show how suitably quantized Ruelle transfer operators can be used to study these fluxes. Among other things, we will discuss:

1 June 2013, room 01, 3pm

Nikolay Tzvetkov (Université de Cergy-Potnoise): Mesures invariantes de type gaussienne pour des EDP hamiltoniennes

Résumé : Nous allons présenter la construction de base pour définir des mesures invariantes de type gaussienne associées à une loi de conservation d'une équation aux dérivées partielles. Ensuite, nous allons donner plusieurs exemples ou cette construction peut être mise en place, en montrant les différentes types de difficultés qui puissent apparaitre.

Carlangelo Liverani (University Roma Tor Vergata, Italy): TBA

29 May 2017, room 05, 3pm

Raphael Lefevere (Université Paris Diderot, Paris, France): TBA

15 May 2017, room 421, 3pm

Freddy Bouchet (ENS de Lyon, France): TBA

24 April 2017, room 05, 3pm

Antonio Lerario (SISSA, Trieste, Italy): TBA

20 March 2017, room 421, 3pm

Massimiliano Gubinelli (Hausdorff Center for Mathematics, Bonn, Germany): Weak universality of fluctuations and singular stochastic PDEs

Abstract: Mesoscopic fluctuations of microscopic (discrete or continuous) dynamics can be described in terms of nonlinear stochastic partial differential equations which are universal: they depend on very few details of the microscopic model. Due to the extreme irregular nature of the random field sample paths, these equations turn out to not be well-posed in any classical analytic sense. In this talk I will review recent progress in the mathematical understanding of such singular equations and of their (weak) universality. If time permits I will discuss the case of the one dimensional Kardar-Parisi-Zhang equation and of the three dimensional Stochasic Allen-Cahn equation.

20 February 2017, room 421, 3pm

Alexandre Boritchev (Institut Camille Jordan, Université Lyon 1, France): Multidimensional Burgers Turbulence

Abstract: The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all models of turbulence. In particular, K41 and corrections to it provide estimates of small-scale quantities such as increments and energy spectrum for a 3D turbulent flow. However, because of the well-known difficulties involved in studying 3D turbulent flows, there are no rigorous results confirming or infirming those predictions.

Here, we consider a model for 3D turbulence: turbulence for the multi-dimensional potential Burgers equation. In the space-periodic case with a stochastic white in time and smooth in space forcing term, we give sharp estimates for small-scale quantities such as increments and energy spectrum and we obtain results on the speed of convergence to the stationary measure.

23 January 2017, room 05, 3pm

Darryl D. Holm (Imperial College London, Mathematics Department, UK): Variational Principles for Stochastic Fluid Dynamics

Abstract: For the purpose of estimating model error in predictions of climate and weather variability, we propose an approach which includes stochastic processes. The idea is to represent unknown errors, as cylindrical noise appearing in systems of stochastic evolutionary PDEs which derive from variational principles that are invariant under a Lie group action. The main objective of the presentation is the inclusion of stochastic processes in ideal fluid dynamics, via a variational principle which is invariant under particle relabelling by smooth invertible maps. Examples include Euler’s fluid equations for incompressible flows and also approximate GFD (Geophysical Fluid Dynamics) equations for ocean and atmosphere circulation.

The approach is via a stochastic extension of the Hamiltons principle for fluid which imposes a constraint of stochastic transport of advected quantities, whose spatial correlations are obtained from observed data for tracer transport. The equations we derive via this approach keep their deterministic form and geometric meaning, which both derive from the variational principle. However, their transport vector field becomes stochastic, corresponding to stochastic Lagrangian particle paths. This means, for example, that Kelvin’s circulation theorem for the stochastically modified Euler equations for incompressible flow has the same integrand as in the deterministic case, but its circulation loop moves with the fluid flow along stochastic Lagrangian paths.

Details and examples for GFD may be found in: D.D. Holm, Variational principles for stochastic fluid dynamics, [2015] Proc Roy Soc A, 471: 20140963.

21 November 2016, room 421, 3pm

Filippo Santambrogio (Université Paris-Sud, France): Une introduction au transport optimal et aux flots de gradient dans l'espace de Wasserstein

Résumé: Le séminaire se composera essentiellement de trois parties. D'abord, pour introduire les équations d'évolution que je traiterai ensuite, je présenterai les généralités sur les flots de gradient dans l'espace Euclidien, sous la forme $x'(t)=-DF(x(t))$, et la manière d'adapter cette analyse à des espaces métriques.

Ensuite, je ferai une petite introduction au transport optimal, avec les problèmes de Monge, de Kantorovich, et la dualité, et aux distances de Wasserstein $W_p$ qu'on peut définir sur l'espace des mesures de probabilité en utilisant la valeur minimale d'un problème de transport.

Je terminerai en regardant quelles équations d'évolution sont en fait des flots de gradient dans l'espace des mesures de probabilité muni de la distance $W_2$, ce qui inclut l'équation de la chaleur, de Fokker-Planck, des milieux poreux..., ainsi que beaucoup d'autres EDP dans la dynamique des populations ou des gaz.

17 Octobre 2016, room 421, 3pm

Anne-Sophie de Suzzoni (Université Paris 13, Sorbonne Paris Cité, France): Invariant measures on the line for a class of Hamiltonian equations

Abstract: In this talk based on a joint work with F. Cacciafesta, I will consider a Hamiltonian equation whose Hamiltonian has a kinetic and a potential part. I will explain how to obtain a weakly invariant measure under the flow of this equation on the real line by applying cut-offs in space and in frequency and by localising the non linearity (or potential part of the Hamiltonian). These cut-offs and localisation reduce the problem to a finite dimensional one. The meaning to "weakly invariant" will be precised. The strategy uses two main ingredients : first, what one might refer to as the Prokhorov-Skorohod technique, which ensures that with uniform in the different cut-offs and localisation estimates on invariant measures on finite dimension, we get a weakly invariant measure on the line by passing to the limit; then, we use the Feynman-Kac theorem to get these estimates.

These ideas apply to a general range of equations. I will give an example with the Schrödinger equation with variable coefficients.

3 Octobre 2016, room 421, 3pm

Laurent Thomann (Université de Lorraine, France): Mesures invariantes pour les équations dispersives

Résumé: On commencera par faire des rappels sur la notion de mesure de probabilité invariante. On verra ensuite des applications à l'étude en temps grand de systèmes dynamiques. Dans un second temps, on présentera l'équation de niveau fondamental de Landau, et on construira des solutions globales à faible régularité à l'aide de mesures invariantes (Gibbs et bruit blanc).

5 September 2016, room 05, 3pm

Alexander Bufetov (Aix-Marseille Université & CNRS, France): Mesures conditionnelles des processus déterminantaux

Abstract: A cause de l'interaction non locale entre des particules, les propriétés dynamiques des mesures déterminantales sont très différentes de celles des mesures de Gibbs: par exemple, pour le sinus-processus, le nombre des particules dans un intervalle est presque surement détermine par la configuration en dehors de l'intervalle (rigidité de Ghosh-Peres).

Le résultat principal de cet expose est une formule explicite pour les mesures conditionnelles dans un intervalle par rapport a la configuration dans le complément de l'intervalle pour une classe assez large des processus déterminantaux en dimension 1. Le rôle clef est joue par la quasi-invariance de ces processus par le groupe de difféomorphismes au support compact. Pour le cas spécial du processus au noyau Gamma, dans le cadre discret, la quasi-invariance par le groupe symétrique infini a été établi par Olshanski; le cas général sera traité dans l'expose qui se base sur deux prépublications.

[1] A.I. Bufetov, Quasi-Symmetries of Determinantal Point Processes.

[2] A.I. Bufetov, Conditional measures of determinantal point processes.

20 June 2016, room 05, 3pm

Thierry Bodineau (École Polytechnique, France): Large deviations and non-equilibrium statistical mechanics

Abstract: In this talk, we will review some results on the steady states of diffusive systems maintained off equilibrium by two heat baths at unequal temperatures. Using the framework of the hydrodynamic limits, we will discuss the large deviations of the heat current through these systems. In particular, we will explain the occurrence of dynamical phase transitions which may occur for some models.

13 June 2016, room 05, 3pm

Noé Cuneo (University of Genève, Switzerland): Non-equilibrium steady states for chains of rotors

Abstract: We study the existence of an invariant measure (called non-equilibrium steady state) for chains of rotors interacting with stochastic heat baths at different temperatures. The interesting issue is that rotors with high energy tend to decouple from their neighbors, due to the fast oscillations of the interaction forces. This phenomenon makes the existence of a steady state challenging to prove, and prevents the system from relaxing exponentially rapidly to its invariant measure. I will introduce the model and recall some results about Lyapunov functions. Then, I will sketch the proofs that we have written with J.-P. Eckmann and C. Poquet for chains of length 3 and 4. Finally, I will explain why we cannot handle longer chains (for now!).

9 May 2016, room 05, 3pm

Konstantin Khanin (University of Toronto, Canada): On global solutions to the random Hamilton-Jacobi equation

Abstract: We shall discuss a problem of existence and uniqueness of global solutions to the random Hamilton-Jacobi equation. While the problem in the spatially periodic setting is well understood by now, the situation in the non-compact case remains largely open.

In this talk we shall discuss both cases, and present partial results and conjectures in the non-compact situation. We shall also discuss a connection with the problem of KPZ universality.

4 April 2016, Amphi Darboux, 3pm

Sergio Simonella (Technische Universität München, Germany): Evolution of marginals in the hard sphere system

Abstract: The problem of the rigorous validity for the Boltzmann kinetic equation knows no other approach than a detailed study of the BBGKY hierarchy, namely the complete set of equations for the evolution of marginals in a large system of interacting particles. Focusing on the paradigmatic case of hard spheres, I will describe the hierarchy and the structure of its solution. Then I will discuss some method for its derivation within minimal regularity assumptions.

14 March 2016, room 05, 3pm

Nicolai Krylov (University of Minnesota, USA): On the existence of ${\bf W}_p^2$ solutions for fully nonlinear elliptic equations under relaxed convexity assumptions

Abstract: We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains without requiring $H$ to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of $H$ at points at which $|D^{2}v|\leq K$, where $K$ is any given constant. For large $|D^{2}v|$ some kind of relaxed convexity assumption with respect to $D^{2}v$ mixed with a VMO condition with respect to $x$ are still imposed. The solutions are sought in Sobolev classes.

16 February 2016, room 314, 4pm

Laszlo Erdös (Institute of Science and Technology, Austria): Universality of spectral statistics of random matrices, Part II

15 February 2016, room 314, 3pm

Laszlo Erdös (Institute of Science and Technology, Austria): Universality of spectral statistics of random matrices, Part I

Abstract: E. Wigner predicted that the local eigenvalue gap statistics of sufficiently complex quantum systems are universal. One prominent model to test this theory is matrices with independent random entries. In the recent years the universality conjecture for a very broad class of random matrices has been resolved. In these talks I will present the main questions and explain the novel analytical techniques that have been developed to solve them.

1 February 2016, room 05, 3pm

Mark Freidlin (University of Maryland, USA): Large-time effects of small perturbations and the simplex of invariant measures

Abstract: Perturbations of dynamical systems and semi-flows will be considered. Long-time evolution of the perturbed dynamics, under some assumptions, consists of a slow component, which is, actually, a motion on the simplex of invariant probability measures of the non-perturbed system, and of a fast component which can be characterized by the invariant measure corresponding to the slow component position. The slow component, in an appropriate time scale, converges weakly to a motion on the simplex. This limiting slow motion can be stochastic even in the case of pure deterministic perturbations of deterministic systems. I will consider various realizations of this general approach for perturbations of systems defined by ODEs and PDEs.

18 January 2016, room 05, 3pm

Shizan Fang (Université de Bourgogne, France): Navier-Stokes equations on Riemannian manifolds

Abstract: On a Riemannian manifold, there exist several "Laplacian operators" on vector fields. In this talk, we will present different probabilistic behaviors behind these equations.

11 January 2016, room 05, 3pm

Isabelle Gallagher (Université Paris-Diderot, France): De la dynamique moléculaire aux équations de l’acoustique et de Stokes-Fourier

Résumé: La question du passage d'une description microscopique de particules (via la mécanique déterministe newtonienne) à une description macroscopique (via des équations de la mécanique des fluides) est un problème largement ouvert. Dans cet exposé nous présenterons quelques progrès récents dans des régimes linéaires, obtenus avec Thierry Bodineau et Laure Saint-Raymond.

14 December 2015, room 201, 3pm

Giambattista Giacomin (Université Paris Diderot, France): Weak noise and non hyperbolic unstable fixed points

Abstract: We consider one dimensional ordinary stochastic differential equations driven by additive Brownian motion with small variance. When the variance is zero such equations have an unstable non-hyperbolic fixed point and the drift near such a point has a power law behavior. For positive variance, the fixed point property disappears, but it is replaced by a random escape or transit time which diverges as the variance tends to zero. We show that this random time, under suitable (easily guessed) rescaling, converges to a limit random variable that (essentially) depends only on the power exponent associated to the fixed point. Such random variables, or laws, have therefore a universal character and they arise of course in a variety of contexts. We obtain quantitative sharp estimates, notably tail properties, on these universal laws. Work in collaboration with M. Merle.

16 Novembre 2015, room 201, 3pm

Juraj Földes (Université Libre de Bruxelles, Belgium): Large parameter limits for stochastically forced equations

Abstract: Due to sensitivity with respect to initial data and parameters, individual solutions of the basic equations of fluid mechanics are unpredictable and seemingly chaotic. However, some of their statistical properties are robust. As early as the 19th century it was conjectured that turbulent flow cannot be solely described by deterministic methods, and indicated that a stochastic framework should be used. In this framework, invariant measures of the stochastic equations of fluid dynamics presumably contain the statistics posited by the basic theories of turbulence.

In this talk we investigate properties of invariant measures for the Boussinesq equations and Magneto-hydrodynamic equations in the presence of a degenerate stochastic forcing acting only in the temperature component. The main goal is to prove convergence of invariant measures in singular limits when Prandtl or Rossby and magnetic Reynolds numbers approach infinity.

More precisely, we show a general framework for converting the problem of convergence of measures to the question of finite time convergence of solutions. Then we analyze singular limit problems in a stochastic setting. This is a joint work with S. Friedlander (U. of Southern California), N. Glatt-Holtz (Virginia Tech), G. Richards (U. of Rochester), and E. Thomann (Oregon State).

19 October 2015, room 05, 3pm

Jani Lukkarinen (University of Helsinki, Finland): Hydrodynamics without scaling limits: Thermalization in harmonic particle chains with velocity flips

Abstract: Hydrodynamics, and the closely related Fourier's law of heat conduction, are successful models for macroscopic transport in many physical systems. Although complete understanding of their microscopic origin is still missing, solid progress has been made in deriving macroscopic and mesoscopic evolution equations using scaling limits. However, the scaling limits sometimes offer only a partial picture of the dynamics behind the transport phenomena. One issue obscured by hydrodynamic scaling limits is "thermalization". This refers to the evolution of an initial state, assumed to be sufficiently chaotic but otherwise fairly unrestricted, into a local equilibrium state which determines the initial data of the hydrodynamic evolution equations. The thermalization process is typically hard to control rigorously as it involves evolution at the microscopic scales, while the hydrodynamic evolution occurs at time scales $O(L^2)$, where $L$ denotes the spatial length scale of the system.

In this talk, I will discuss thermalization in a particle chain where the particles interact via a harmonic potential and each particle flips the direction of its velocity randomly. It turns out that after a relatively short time, the average kinetic temperature profile satisfies the Fourier's law, in a local microscopic sense, without assuming that the initial data is close to a local equilibrium state. The bounds derived here prove that the thermalization period is at most of the order of $L^{2/3}$ where L denotes the number of particles in the chain. In particular, even before the diffusive time scale, Fourier's law becomes a valid approximation of the evolution of the kinetic temperature profile. The talk is mainly based on J. Stat. Phys. 155 (2014) 1143-1177 (arxiv.org/abs/1308.4901).

5 October 2015, room 05, 3pm

Vahagn Nersesyan (Université de Versailles, France): The Gallavotti-Cohen principle for randomly forced PDE's

Abstract: The mathematical, physical, and numerical aspects of the Gallavotti-Cohen principle are widely studied in the literature. The previous rigorous mathematical works are mainly concerned with finite-dimensional stochastic systems. The aim of this talk is to present a result for an infinite-dimensional model provided by the randomly forced Burgers equation. Assuming that the force is rough with respect to the space variables and has a non-degenerate law, we prove a large deviations principle (LDP) for some unbounded functionals. Then in the case of the entropy production functional, we show that the rate function of the LDP satisfies a symmetry of Gallavotti-Cohen type. This is a joint work with V. Jaksic, C.-A. Pillet and A. Shirikyan.

15 June 2015, room 421, 3pm

Vojkan Jaksic (McGill University, Canada): Physics and mathematics of fluctuation relations

Abstract: The discovery of fluctuation relations revolutionized our understanding of non-equilibrium statistical mechanics. In this talk, I will try to describe their physical and mathematical significance in simplest possible terms.

8 June 2015, room 421, 3pm

Stanislav Molchanov (University of North Carolina at Charlotte, USA): Anderson parabolic problem with the random non-stationary potentials, localization versus intermittency (review)

Abstract: a) Stationary potentials, relations with the Anderson localization in the solid state physics. Quenched and annealed asymptotics for the moments (by examples).

b) Non-stationary potentials ergodic in space and short correlated in time. Magnetic field in such (turbulent) velocity fields (dynamic problem). Intermittency.

c) Population dynamics. KPP model with propagating front. Intermittency inside the front. Instability of the steady states with respect to the small random perturbations.

d) Open problems.

18 May 2015, room 421, 3pm

Olivier Glass (Université Paris-Dauphine): Contrôlabilité pour l'équation d'Euler non isentropique

Résumé: On considère la question de contrôlabilité de l'équation d'Euler non isentropique pour les gaz compressibles polytropiques, dans le contexte de solutions faibles d'entropie à petite variation totale. On considère le système à la fois en variables lagrangiennes et en variables euleriennes, et on obtient un résultat de contrôlabilité frontière dans les deux cas.

13 April 2015, room 421, 3pm

Vahagn Nersesyan (Université de Versailles): Local large deviations principle for occupation measures of the damped nonlinear wave equation perturbed by a white noise

Abstract: In this talk, we will consider the damped nonlinear wave (NLW) equation driven by a spatially regular white noise. Assuming that the noise is non-degenerate in all Fourier modes, we will establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation. The proof relies on a generalization of methods developed in [JNPS1] and [JNPS2] in the context of kick forced dissipative PDE’s with parabolic regularization property such as, for example, the Navier–Stokes system or the complex Ginzburg–Landau equations. We show that a high concentration towards the stationary measure is impossible, by proving that the rate function that governs the LDP cannot have the trivial form (i.e., vanish on the stationary measure and be infinite elsewhere). This is a joint work with D. Martirosyan.

[JNPS1] V. Jaksic, V. Nersesyan, C.-A. Pillet, A. Shirikyan, Large deviations from a stationary measure for a class of dissipative PDE’s with random kicks, Comm. Pure Appl. Math. 68 (2015).

[JNPS2] V. Jaksic, V. Nersesyan, C.-A. Pillet, A. Shirikyan, Large deviations and mixing for dissipative PDE’s with unbounded random kicks, preprint, 2014.

23 March 2015, room 421, 3pm

Elena Kartashova (Johannes Kepler University, Linz, Austria): Wave turbulence theory: historical overview and open questions

Abstract: The beginning of the wave turbulence theory (WTT) is considered to be 1967, when it was first discovered that the so-called wave kinetic equation has stationary solutions. Stationary solutions, given by power law functions, describes the energy spectrum of a weakly nonlinear wave system in the Fourier space. The search for such spectra and the study of their properties is the subject of kinetic WTT. The next important achievement in this area relates to 1990, when it was discovered the existence of independent resonant clusters. The problem of constructing of the resonance cluster set for given dispersive wave system can be reduced to solving of a Diophantine equation in several variables in high powers. The construction of resonance clusters and the study of their properties is the subject of discrete WTT.

16 March 2015, Lecture theater Hermite, 4pm

Herbert Spohn (Zentrum Mathematik, TU München): The noisy Burgers equation with several components

Abstract: My interest is the noisy Burgers equation on the entire real line, the noise being space-time white noise. In particular I would like to understand the structure of the stationary covariance. I recall the exact solution in the scalar case. The case of several components is of great physical interest with very few mathematical results only. Conjectures and preliminary results will be discussed.

9 March 2015, room 421, 3pm

Jürg Fröhlich (ETH - Institute for Theoretical Physics): Effective dynamics in quantum theory

Abstract: I will discuss some important examples of effective (stochastic) dynamics in quantum theory:

(1) A model of a quantum particle coupled to a heat bath at positive temperature; the theorem being that the particle exhibits Quantum Brownian Motion (work with De Roeck and Pizzo);

(2) A simplified treatment of (1); but with a random external potential added (Anderson model coupled to a heat bath); the theorem being that thermal noise destroys Anderson localization (work with Schenker);

(3) A mesoscopic quantum system subjected to repeated projective measurements of a single "observable"; our results concerning a theory of "events and quantum jump processes" (work with Ballesteros, Fraas and Schubnel).

9 February 2015, room 421, 3pm

Andrey Dymov (Université de Cergy-Pontoise): Nonequilibrium statistical mechanics of solids in medium

Abstract: Investigation of the energy transport in solids is one of the main problems in the nonequilibrium statistical mechanics. Since it turns out to be extremely difficult, usually one studies toy models, possessing additional ergodic properties. A common idea is to consider a Hamiltonian system of particles where each mode is a subject to stochastic perturbation. Clearly, it is important to study the case when the perturbation goes to zero.

In this talk I will discuss dynamics of an anharmonic system of weakly interacting oscillators, where each oscillator is weakly coupled with its own stochastic Langevin thermostat. The system can be interpreted as a solid plugged in medium and weakly interacting with it. I will prove that, under the limit when the couplings of oscillators with each other and with the thermostats go to zero with some precise scaling, behaviour of the system is governed by an effective equation which is a rather nice dissipative SDE. I will show that under the limit above, dynamics of the energy satisfies laws, which resemble the Fourier law and the Green-Kubo formula (but which are not the F. law and the G.-K. formula).

1 December 2014, room 421, 3pm

François Golse (Ecole polytechnique, Paris): Mean field limits and kinetic models

Abstract: Various kinetic models, such as the Vlasov-Poisson system, are formally derived from the dynamics of a large number of interacting particles in some scaling limit known as the mean field limit. There are two different approaches to this type of limit: one involves the notion of empirical measure of the particle system in the single-particle phase space, while the other is based on a procedure known as the BBGKY hierarchy.

The first part of the talk (work in collaboration with C. Mouhot and V. Ricci) explains how both approaches are related, and discusses convergence rates based on the work of R.L. Dobrushin (Func. Anal. 1979). The second part of the talk explains how Dobrushin's estimate can be extended to a variant of the Vlasov-Maxwell system.

17 November 2014, room 421, 3pm

Marcello Porta (University of Zürich): Mean-field evolution of fermionic systems

Abstract: In this talk I will discuss the dynamics of interacting fermionic systems in the mean-field regime. Compared to the bosonic case, fermionic mean-field scaling is naturally coupled with a semiclassical scaling, making the analysis more involved. From a physical point of view, as the number of particles grows one expects the quantum evolution of the system to be effectively described by Hartree-Fock theory. The next degree of approximation is provided by a classical effective dynamics, corresponding to the Vlasov equation.

I will consider initial data which are close to quasi-free states, both at zero and at positive temperature, with an appropriate semiclassical structure. Under some regularity assumptions on the interaction potential I will show that the time evolution of such initial data stays close to a quasi-free state, with reduced one-particle density matrix given by the solution of the time-dependent Hartree-Fock equation. The result holds for all (semiclassical) times, and gives effective bounds on the rate of convergence towards the Hartree-Fock dynamics as the number of particles goes to infinity.

20 October 2014, room 421, 3pm

Viviane Baladi (ENS, Paris): The spectrum of Sinai billiard flows

Abstract: Sinai billiard maps in dimension two have been known to be exponentially mixing (L.-S. Young) for almost two decades, and recent work of Demers and Zhang have shed new light on the spectrum of their transfer operators. The situation for the continuous time Sinai billiard is more delicate. I will present recent results and ongoing work on their spectrum. (Joint work with M. Demers and C. Liverani).

6 October 2014, room 421, 3pm

Frédéric Rousset (Université de Paris-Sud, Orsay): L’amortissement Landau pour un modèle simple de particules en interaction

Résumé : Le but de l’exposé sera de présenter, pour le modèle très simple de Vlasov-HMF qui décrit des particules sur le cercle interagissant avec un potentiel régulier, des résultats d’amortissement Landau dans des espaces de Sobolev. Il s’agit de décrire en temps grand pour une équation hamiltonienne et réversible le comportement des solutions qui sont des petites perturbations d’équilibres spatialement homogènes stables. Travail en commun avec Erwan Faou.

12 May 2014, room 421, 3pm

Sergey Nazarenko (University of Warwick): Open questions in Wave Turbulence theory

Abstract: I will outline the main steps in the Wave turbulence approach and highlight some open questions for mathematicians.

7 April 2014, room 421, 3pm

Wei-Min Wang (Université de Cergy-Pontoise): Witten Laplacian and Stochastic NLS

Abstract: We shall discuss some recent works on stochastic NLS on the torus, which proves exponential approach to equilibrium. The most precise results are obtained on the circle, both for the focusing and the defocusing cases. (These are joint works with Carlen, Froehlich and Lebowitz in various combinations.)

24 March 2014, Amphithéâtre Hermite, 3pm

Nicolas Burq (Université de Paris-Sud, Orsay): Gibbs measures and NLS on planar domains

Abstract: In this talk I will present the construction of Gibbs measures and Wick reordered NLS on bounded planar domains (or more generally any compact surface with boundary). I will also show how these measures allow to construct a weak flow for NLS, and show some perspectives toward strong flows. This is a joint work with L. Thomann and N. Tzvetkov.

10 February 2014, room 421, 3pm

Thierry Bodineau (École Polytechnique, Palaiseau): Diffusion pour une particule marquée dans un gaz dilué de sphères dures

Résumé : On étudie le mouvement d’une particule marquée dans un gaz dilué de sphères dures à l’équilibre. Après changement d’échelle de l’espace et du temps, on montre que cette particule suit un mouvement brownien. (travail commun avec I. Gallagher et L. Saint-Raymond)

20 January 2014, room 05, 3pm

Yves Le Jan (Université de Paris-Sud, Orsay): Lacets markoviens

Résumé : On présentera quelques résultats anciens et nouveaux sur les ensembles Poissoniens de lacets markoviens, leurs amas, et leurs relations avec le champ libre.

16 December 2013, room 421, 3pm

Denis Bernard (LPT-ENS, Paris): On the relation between SLE and CFT (or, more generally, statistical mechanics)

Abstract: Stochastic Schramm Loewner Evolutions (SLE) are Markov processes describing fractal curves or interfaces in two-dimensional critical systems. After a presentation of the objects SLE deals with, and of the basics tools involved in their description, I will present the essential points underlying the connection between statistical mechanics and processes which, in the present context, leads to a connection between SLE and conformal field theory (CFT). The lecture is intended to be at an introductory level.

2 December 2013, room 421, 3pm

Alexandre Boritchev (Université de Genève): Hyperbolicité des minimiseurs pour l'équation de Burgers stochastique

Résumé : Nous regardons l'équation de Burgers stochastique du point de vue lagrangien. En d'autres mots, nous étudions le comportement dynamique des minimiseurs d'énergie qui induisent la description variationnelle des solutions. Sous des conditions de non-dégénérescence sur le forçage aléatoire, nous prouvons l'hyperbolicité de ces minimiseurs. Nous simplifions considérablement la preuve donnée dans l'article [E, Khanin, Mazel, Sinai, Annals, 2000]. Nous conclurons par le lien entre ce problème et celui de la convergence vers la mesure stationnaire pour les solutions de l'équation.

Travail en collaboration avec K. Khanin (Université de Toronto)

18 November 2013, room 421, 3pm

Armen Shirikyan (University of Cergy-Pontoise): Large deviations from a stationary measure for dissipative PDE's with random kicks

Abstract: We study a class of dissipative PDE's perturbed by a random kick force. It is well known that if the random perturbation is sufficiently non-degenerate, then the Markov process associated with the problem in question has a unique stationary distribution, which is exponentially mixing. In addition, the strong law of large numbers and the central limit theorem hold and give the large-time behaviour of probabilities for small deviations of the time average of continuous functionals from their spatial average with respect to the stationary distribution. In this talk, I discuss the asymptotics of probabilities of order-one deviations from the stationary measure. Our main result shows that the occupation measures of solutions satisfy the LDP with a good rate function. The proof is based on Kifer's criterium for LDP, a Lyapunov-Schmidt type reduction, and a general result on long-time behaviour of generalised Markov semigroups. The result applies to the 2D Navier-Stokes system, Ginzburg-Landau equation, and other dissipative PDE's. This is a joint work with V. Jaksic, V. Nersesyan, and C.-A. Pillet.

4 November 2013, room 421, 3pm

Massimiliano Gubinelli (University Paris-Dauphine): SPDEs, paraproducts and all that

Abstract: In this talk we explain recent advances in understanding of the functional analytic structure of solutions to non-linear SPDEs and their application to the study of various model of mathematical physics: a 2d parabolic Anderson model, the 3d stochastic quantisation equation and the 1d Kardar-Parisi-Zhang equation. These advances have been possible thanks to a generalisation of the theory of controlled rough paths. In particular we discuss the role of multiscale decomposition of distributions and of the notion of paraproduct in the analysis of this problem. I will assume only basic knowledge of functional analysis, in particular no results of rough path theory or stochastic analysis are needed to follow the talk.

14 October 2013, room 01, 3pm

Tomasz Komorowski (Polish Academy, Warsaw): Long time energy transfer in the random Schrodinger equation

Abstract: We consider the long time behavior of solutions of the d-dimensional linear Boltzmann equation that arises in the weak coupling limit for the Schrodinger equation with a time-dependent random potential. We show that the intermediate mesoscopic time limit satisfies a Fokker-Planck type equation with the wave vector performing a Brownian motion on the (d − 1)-dimensional sphere of constant energy, as in the case of a time-independent Schrodinger equation. However, the long time limit of the solution with an isotropic initial data satisfies an equation corresponding to the energy being the square root of a Bessel process of dimension d/2. This is a joint work with Lenya Ryzhik.

30 September 2013, room 05, 3pm

Stefano Olla (University Paris-Dauphine): From dynamics to thermodynamics

Abstract: One of the purposes of statistical mechanics is to explain thermodynamics in terms of ‘microscopic’ dynamics. The general consensus is that thermodynamics describe the ‘macroscopic’ behaviour of some quantities (energy, pressure, …) under the influence of external forces or thermostats. Macroscopic means that laws of thermodynamics emerge for ‘large’ systems and in a large time scale.

I will present a precise mathematical approach where thermodynamics laws (Carnot cycles, Kelvin and Clausius principles, etc.) are obtained through space-time scaling limits. In this approach, that I consider ultraorthodox, the problem of thermodynamics, and in particular its second principle, is reduced to a purely mathematical problem, very difficult indeed.

Avoiding a general theory, I will talk about the simplest system: a ‘one dimensional’ rubber subject to tension and heat bath. This will be modelled microscopically by a (classical) chain of oscillators (of Fermi-Pasta-Ulam type) subject to boundary forces and eventually stochastic Langevin thermostats.

Irreversible isothermal and adiabatic transformations can be obtained for the evolution of ‘local’ energy and stress, by hydrodynamic limits, performing space-time scalings. Quasi-static reversible thermodynamic transformations, object of the classical thermodynamics, are then obtained by further time rescaling.

From the mathematical point of view, isothermal transformation can be fully proven, while adiabatic transformations are the main challenge.

16 September 2013, room 01, 3pm

Alberto Maiocchi (University of Cergy-Pontoise): Adiabatic invariants in the thermodynamic limit

Abstract: I will show how to construct an adiabatic invariant for a large 1-d lattice of particles, such that the time evolution of this quantity is negligible during exponentially long times as the temperature goes to zero. In difference with the results available in the literature, the present result holds uniformly in the thermodynamic limit, when the number of degrees of freedom tends to infinity, while the temperature stays fixed.

In the first part of the talk I will present an introduction to the problem and state the main result, stressing the consequences for physics and giving a sketch of the proof. The second part will be devoted to the illustration of the techniques used to control the decay of the spatial correlations with respect to the Gibbs measure, as they play a key role in the proof. This techniques is an adaptation of the Dobrushin method.

1 July 2013, room 01, 3pm

Arnaud Debussche (ENS Cachan - Bretagne): Comportement en temps long pour les lois de conservation stochastiques

Résumé : Il s'agit d'un travail en commun avec J. Vovelle dans lequel nous étudions le comportement en temps long des solutions de lois de conservations stochastiques. L'existence de celles-ci a été obtenue par plusieurs auteurs. Nous avons obtenue un résultat très général grace à une généralisation au cadre stochastique de la formulation cinétique introduite par Lions, Perthame et Tadmor. Cette formulation est très puissante car elle permet de garder trace de la dissipation. En utilisant un lemme de moyenne, nous parvenons à montrer que, si l'equation n'est pas dégénérée, la dissipation d'énergie est suffisante pour assurer l'existence d'une mesure invariante. De plus, en dimension un et pour des flux au plus quadratique nous montrons qu'il y a unicité de la mesure invariante, et donc ergodicité. Nous généralisons ainsi un résultat de E, Khanin, Mazel et Sinai, obtenu pour l'équation de Burgers, à des équations générales pour lesquelles la formule de Hopf-Lax-Oleinik n'est pas valide.

17 June 2013, room 05, 3pm

Vojkan Jaksic (University of McGill): Non-equilibrium statistical mechanics of the spin-boson model

Abstract: The non-equilibrium spin-boson model describes interaction of a quantum dot (say, spin 1/2) with several independent bosonic thermal reservoirs which are initially in thermal equilibrium at distinct temperatures. The temperature differentials result in non-trivial energy/entropy flux across the system. In this talk we will show how suitably quantized Ruelle transfer operators can be used to study these fluxes. Among other things, we will discuss:

- The large deviation principle for the Full Counting Statistics associated to the repeated measurement protocol of the energy flow and resulting fluctuation relations/theorems.
- The linear response theory and the Fluctuation-Dissipation Theorem (Green-Kubo formula, Onsager reciprocity relation, and Central Limit Theorem for the heat fluxes).
- The relaxation to non-equilibrium steady state.
- Non-equilibrium statistical mechanics of the model in the Van
Hove Limit. One of our goals is to give a unified treatment of these
topic via study of the spectral resonances of quantum Ruelle transfer
operators.

1 June 2013, room 01, 3pm

Nikolay Tzvetkov (Université de Cergy-Potnoise): Mesures invariantes de type gaussienne pour des EDP hamiltoniennes

Résumé : Nous allons présenter la construction de base pour définir des mesures invariantes de type gaussienne associées à une loi de conservation d'une équation aux dérivées partielles. Ensuite, nous allons donner plusieurs exemples ou cette construction peut être mise en place, en montrant les différentes types de difficultés qui puissent apparaitre.