Bernou Armand : The Cercignani-Lampis boundary condition in kinetic theory.

This talk focuses on various applications of a deterministic Harris subgeometric theorem to study the convergence towards the steady state in kinetic theory with boundary condition. We start with the already difficult case of the free-transport equation (particles go in straight line without interacting with each other) inside a domain with a diffuse reflection at the boundary: in this case, when the particle hits the wall, a new velocity is chosen at random, independent of the incoming one. This boundary condition has been heavily studied in the last fifteen years. Then, we will focus on a generalization used by physicists: the Cercignani-Lampis boundary condition, and give some details about the added difficulty. If time allows, we will briefly mention some related results in collisional kinetic theory.

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Cavallazzi Thomas : Weak propagation of chaos for Lévy-driven McKean-Vlasov SDEs.

We will deal with McKean-Vlasov stochastic differential equations driven by a $\alpha$-stable process with $\alpha >1$ under Hölder-type assumptions on the drift coefficient. We will study the associated semi-group acting on functions defined on the space of measures through the related backward Kolmogorov PDE stated on the space of measures. We will focus in particular on its regularizing properties. The study relies on differential calculus in the space of measures and on Itô's formula along the flow of measures generated by a jump process. We will finally use this to prove quantitative weak propagation of chaos estimates for the associated interacting particle system.

Colombani Laetitia : Propagation of chaos in a network of FitzHugh-Nagumo.

FitzHugh-Nagumo equations have been suggested in 1961 to model neurons. Stochastic versions of these equations have since been developed. The specificity of these SDE is a cubic term in the drift, which needs us to pay attention. With Pierre Le Bris, we have studied the behavior of a network on N neurons, interacting with each other, when N tends to infinity. We prove an uniform in time propagation of chaos in a mean-field framework, with a coupling method suggested by Eberle (2016). During this talk, I will present this model and the idea of the method.

Costa Manon : Portfolio optimization under CV@R constraint with stochastic mirror descent

In this talk I will present a problem of optimal portfolio allocation with CV@R constraints when dealing with imperfectly simulated  financial assets. We use a stochastic biased Mirror Descent to find optimal resource allocation for a portfolio whose underlying assets cannot be generated exactly and may only be approximated with a numerical scheme that satisfies suitable error bounds, under a risk management constraint. We establish almost sure asymptotic properties as well as the rate of convergence for the averaged algorithm.This is joint work with Sébastien Gadat and Lorick Huang.

Le Bris Pierre : Some recent results in propagation of chaos.

Consider a system of N particles interacting with one another. We are interested in the limit, as N goes to infinity, of this particle system, trying to derive from a microscopic point of view (i.e the particle dynamics) a mesoscopic point of view (i.e a statistical description of the system). The notion of propagation of chaos refers to the phenomenon according to which, as N grows in the particle system, two given particles become « more and more » statistically independent. The goal of this talk is to show this phenomenon for various types of particle systems (depending on time) : either in a kinetic framework, or with singular interactions (2D vortex model or 1D log and Riesz gases), or with incomplete interactions (i.e in a graph). In the process, we will describe some current methods used in this field, and focus in particular on proving the convergence in N uniformly in time. This is based on joint work with A. Guillin and P. Monmarché, also mentioning some results with L. Colombani and C. Poquet.

Puel Marjolaine : Fractional diffusion for kinetic equations.

Kinetic equations are used to model gaz particles in particular when they collide. They involve a large number of variable and for that reason, a mathematical analysis of those equations were developed to approximate the solution of the initial kinetic equation by the solution of diffusion equation multiplied by a velocity profile, the equilibrium of the collision operator. The case where the equilibrium is a Gaussian is well known. We are interested in the case where the equilibrium is a heavy tail distribution. Since the pioneer result of Mischler, Mellet, Mouhot, we know that the diffusion equation obtained at the limiting equation for the density is a fractional diffusion equation. The goal of this talk is to present the case of the Fokker Planck equation for which a spectral method provides the result. I will present a « à la Koch » method that seems to be promising method to handle more general equations. This is joint works with Dahmane Dechicha and Gilles Lebeau.

Salez Justin : An entropy-curvature approach to the cutoff phenomenon.

Discovered by Aldous, Diaconis and Shahshahani in the context of card shuffling, the cutoff phenomenon is a remarkable phase transition in the convergence to equilibrium of certain Markov chains. Despite the accumulation of several examples, a general theory is still missing, and identifying the exact mechanisms underlying this phenomenon constitutes one of the most fundamental problems in the area of mixing times. After a brief introduction to this question, I will present a new approach based on entropy and curvature, and use it to establish cutoff for a broad class of Markov chains. Examples include random walks on almost all Abelian Cayley graphs with logarithmic degrees.

Simon Marielle : Slow-fast dynamics and noise-induced periodic behaviors for mean-field excitable systems.

We will study non-linear Fokker-Planck equations describing the infinite population limit of interacting excitable particles subject to noise. Taking a slow-fast dynamics limit, we will describe the emergence of periodic behaviors induced by the noise and the interaction. We will consider in particular the case in which each particle evolves according to the FitzHugh Nagumo model. We will then study the longtime fluctuations of the particles system.

Simonov Nikita : Fast diffusion equations, tails and convergence rates.

Understanding the intermediate asymptotic and computing convergence rates towards equilibria are among the major problems in the study of parabolic equations. Convergence rates depend on the tail behaviour of solutions. This observation raised the following question: how can we understand the tail behaviour of solutions from the tail behaviour of the initial datum ? In this talk, I will discuss the asymptotic behaviour of solutions to the fast diffusion equation. It is well known that non-negative solutions behave for large times as the Barenblatt (or fundamental) solution, which has an explicit expression. In this setting, I will introduce the Global Harnack Principle (GHP), precise global pointwise upper and lower estimates of non-negative solutions in terms of the Barenblatt profile. I will characterize the maximal (hence optimal) class of initial data such that the GHP holds by means of an integral tail condition. As a consequence, I will provide rates of convergence towards the Barenblatt profile in entropy and in stronger norms such as the uniform relative error.

Tardy Yoann : Collisions of the supercritical Keller-Segel particle system.

We study a particle system naturally associated to the $2$-dimensional Keller-Segel equation. It consists of $N$ Brownian particles in the plane, interacting through a binary attraction in $\theta/(Nr)$, where $r$ stands for the distance between two particles. When the intensity $\theta$ of this attraction is greater than $2$, this particle system explodes in finite time. We assume that $N>3\theta$ and study in details what happens near explosion. There are two slightly different scenarios, depending on the values of $N$ and $\theta$, here is one: at explosion, a cluster consisting of precisely $k_0$ particles emerges, for some deterministic $k_0\geq 7$ depending on $N$ and $\theta$. Just before explosion, there are infinitely many $(k_0-1)$-ary collisions. There are also infinitely many $(k_0-2)$-ary collisions before each $(k_0-1)$-ary collision. And there are infinitely many binary collisions before each $(k_0-2)$-ary collision. Finally, collisions of subsets of $3,\dots,k_0-3$ particles never occur. The other scenario is similar except that there are no $(k_0-2)$-ary collisions.