# Program

Thursday September 24th

09:00-09:40 Stanislav Molchanov
09:45-10:25 Thierry Bodineau

11:00-11:40 Jean François Jabir
11:45-12:25 Christophe Profeta

14:45-15:30 Antoine Lejay

16:00-16:40 Gilles Pagès
16:45-17:25 Patrick Cattiaux
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Friday September 25th

09:00-09:40 Fabien Panloup
09:45-10:25 Enrico Piola

11:00-11:40 Ugo Boscain
11:45-12:25 Djalil Chafai

14:00-14:40 François Delarue
14:45-15:30 Stéphane Menozzi

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Vlad Bally (Université Paris-Est, Marne la Vallée)
Thierry Bodineau (CMAP, Ecole Polytechnique)
Ugo Boscain (CMAP, Ecole Polytechnique)
Patrick Cattiaux (Université de Toulouse)
Djalil Chafai (Université Paris-Dauphine)
François Delarue (Université de Nice)
Stéphane Menozzi(Université d’Evry et HSE Moscou
Stanislav Molchanov (Universite de Charlotte et HSE Moscou)
Gilles Pagès (Université Pierre et Marie Curie)
Fabien Panloup (Université de Toulouse)
Enrico Priola (Université de Turin)

V. Bally

“Approximation of Markov semigroups in total variation distance” (Vlad Bally and Clément Rey)

The first goal of this paper is to prove that, regularization properties of a Markov semigroup enable to prove convergence in total variation distance for approximation schemes for the semigroup. Moreover, using an interpolation argument we obtain estimates for the error in distribution sense (at the level of the densities of the semigroup with respect to the Lebesgue measure). In a second step, we build an abstract Malliavin calculus based on a splitting procedure, which turns out to be the suited instrument in order to prove the above mentioned regularization properties. Finally, we use these results in order to estimate the error in total variation distance for the Ninomiya Victoir scheme (which is an approximation scheme, of order 2, for diffusion processes).

T. Bodineau

“Large time asymptotics of small perturbations of a deterministic dynamic of hard spheres”

We consider a tagged particle in a diluted gas of hard spheres. Starting from the hamiltonian dynamics of particles in the Boltzmann-Grad limit, we will show that the tagged particle follows a Brownian motion after an appropriate rescaling. We will also consider a different type of perturbation and relate it to the linearized Boltzmann equation.Joint work with I. Gallagher, L. Saint-Raymond

U.Boscain

“Heat-kernels and intrinsic random walks in Riemannian and sub-Riemannian geometry”

On a sub-Riemannian manifold we define two type of Laplacians: the macroscopic Laplacian as the divergence of the horizontal gradient,once a volume is fixed, and the microscopic Laplacian, as the operator associated with a geodesic random walk. We consider a general class of random walks, where all sub-Riemannian geodesics are taken into account.

The main problems are how to choose an intrinsic volume on the manifold to define the macroscopic Laplacian and how to define a “uniform” probability measure on the set of geodesics starting from a point.

P. Cattiaux

“Non parametric estimation for hypoelliptic kinetic SDE”

D. Chafai

“Autour des gaz de Coulomb”

Cet exposé est centré autour des gaz de Coulomb, notamment en liaison avec des modèles de matrices aléatoires. Il est conçu pour être accessible.

J-F Jabir

“Diffusion processes with conditioned distributions

The subject of this talk concerns the construction of diffusion processes whose time marginal distributions are submitted to satisfy some constraint. After a short presentation of this type of processes and some examples appearing in the literature, I will expose a particular study focused on  time marginal constraints and on its construction by means of penalized approximations. This work is supported by the Chilean Fondecyt Iniciacion project Nº11130705.

Antoine Lejay

« Estimation du paramètre d’un mouvement brownien biaisé »

Le mouvement brownien biaisé est maintenant reconnu comme un processus stochastique important en modélisation et simulation de diffusion dans des milieux présentant des interfaces. Nous traiterons dans cet exposé du problème de l’estimation paramétrique du paramètre de biais qui régit sa dynamique. En particulier, nous montrerons que le cadre simplifié des marches aléatoires biaisées illustre les particularités de ce problème d’estimation par rapport aux approches classiques d’estimation des paramètres d’équationsdifférentielles stochastiques.

S. Menozzi

“Sensitivities of Densities for Diffusions and Markov Chains”

Applications to the Weak Error Analysis in a context of low regularity.We are interested in studying the sensitivity of diffusion processes or their approximations by Markov Chains with respect to a perturbation of the coefficients. As an application, we give a first order expansion for the difference of the densities of a diffusion with Holder coefficients and its approximation by the Euler scheme. Other extensions will be considered. This is a joint work with V. Konakov and A. Kozhina (HSE, Moscow).

S. Molchanov

“Ergodic states in the population dynamics”

G. Pagès

“Statistical confluence of duplicated diffusions and application to Langevin Monte Carlo” Joint work with V. Lemaire and F. Panloup.

We address the following question: given an ergodic Brownian diffusion with a unique invariant distribution, what are the invariant distributions of the duplicated system consisting of two trajectories starting from two different points? We mainly focus on the interesting case where the two trajectories are driven by the same Brownian path. In that framework, we first show that uniqueness of the invariant distribution (weak confluence) of the duplicated system is essentially always true in the one-dimensional case. In the multidimensional case, after exhibiting explicit counter-examples, we provide a series of criterions (of integral type) hich ensure the a.s. and the statistical confluence criterions and also of \textit{a.s. pathwise confluence}. We finally establish that the weak confluence property is connected with an optimal transport problem whose cost function is a non-infinitesimal Lyapunov exponent involving the drift and the diffusion coefficient. We apply our criterions to some   classes of gradient systems with non-convex potentials or diffusions where the confluence is generated by the diffusive component.   As a main application, we apply our results to the optimization of the Richardson-Romberg extrapolation for the numerical approximation by Langevin Monte Carlo of the invariant measure of an ergodic Brownian diffusion

F. Panloup

“On the rate of convergence to equilibrium for fractional SDEs”

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with multiplicative noise component $sigma$. When $sigma$ is constant , it was proved by Hairer that, under some mean reverting assumptions, such a process converges to its equilibrium with a fractional rate (depending on $H$). In this talk, I will focus on some recent extensions to the multiplicative case when $H>1/2$ and in the rougher setting $Hin(1/3,1/2)$. We will particularly insist on the Lyapunov strategy and on the coupling methods developed in this non-Markovian setting. This presentation is based on some collaborations with A. Deya, J. Fontbona and S. Tindel.

E. Priola

“Some uniqueness results for SDEs with jumps and Holder continuous drift term”

We present some recent uniqueness results on additive stochastic differential equations with jumps and Holder continuous and bounded drift term. We concentrate especially on SDEs driven by non-degenerate $\alpha$-stable type Levy preocesses.