# Workshop “Singular Mckean-Vlasov Dynamics” Inria Sophia 14-15 septembre 2017 – exposés

14 SEPTEMBRE

10h – 10h40     : S. Veretennikov

TBA

10h40 – 11h20 : P-E. Chaudru de Raynal

“Regularization by noise for a class of McKean-Vlasov SDEs”

In a seminal work of Zvonkin in 1974, it is shown that in_nitesimal stochastic perturbation can restore uniqueness for ordinary di_erential system. In this talk, we show that this phenomenon still occurs for a class of McKean-Vlasov SDEs with rough drift in space and measure arguments. This last result could seems unexpected at _rst sight, since the noise only acts on the space variable. We will show that some particular structural assumptions on the system allow us to recover this phenomenon.Our proof, in the spirit of Zvonkin Work, consists in exhibiting suitable smoothing properties of the Partial Di_erential Equation associated to the generator of the Markov Process. This leads us to study a PDE set on the Cartesian product of the real and the probability measure on the real line, where the derivative along the measure is understood in the Lions sense. Especially, smoothing e_ect on the measure direction are exhibited

11h20 – 12h     : J-F. Jabir

“Lagrangian Stochastic Models with Specular Boundary Conditions”

14h – 14h40     : O. Faugeras

“On a certain McKean-Vlasov equation that arises in neuroscience”

I introduce the problem of characterizing the thermodynamic limit of networks of Hopfield neurons, also known as convolutional networks. I then show existence and uniqueness of the limit McKean-Vlasov equations using a method inspired from the theory of deterministic Volterra equations. This result may be usefulfor shedding some light on
the reasons why convolutional networks work so well and for numerical experiments.

14h40 – 15h20 : N. Champagnat

« Convergence to the minimal quasi-stationary distribution for Fleming-Viot particle systems”

This is joint work with Denis Villemonais (Univ. Lorraine)We consider a general Markov process almost surely absorbed in finite time in a cemetery point. The first goal of the talk is to present general criteria based on Lyapunov functions ensuring the convergence of conditional distributions of the process given its non-absorption to a quasi-stationary distribution. The convergence is in total variation, exponential and with a pre-factor depending on the initial distribution. In particular, these criteria cover cases where there is no uniqueness of quasi-stationary distributions. We then consider the associated particle system, with a mean-field interaction taking place only when one of the particles is absorbed. This particle system is known as Fleming-Viot process. We prove that its invariant measure converges when the number of particles converges to infinity to the so-called minimal QSD. This was conjectured to be a general principle in cases where there is no uniqueness of the quasi-stationary distribution, but it was known only in few cases.

15h40 – 16h20 : S. Roelly

“Statistical mechanics approach for non regular infinite-dimensional SDEs I & II”

In these talks we construct and represent the solution of an infinite-dimensional SDE of the formdX_i(t) = b_t(\theta_i  X) dt + dB_i(t) , i\in N,as Gibbs equilibrium state on a path space, which solves a variational principle.This general statistical mechanics approach could also be used in a framework of propagation of chaos.

16h20 – 17h : D. Dereudre

“Statistical mechanics approach for infinite dimensional SDE”

In this talk we will investigate representations of infinite dimensional SDE via some statistical mechanics concepts as Gibbs equilibrium states and variational principle. This approach could allow to build infinite dimensional SDE related to the propagation of chaos.

15 SEPTEMBRE

9h30 – 10h10  : B. Jourdain

“Existence to a calibrated regime-switching local volatility model”

By Gyongy’s theorem, a local and stochastic volatility (LSV) model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented by Guyon and Henry-Labord\`ere (2011), provide an efficient calibration procedure. But so far, no global existence result is available for the limiting SDE. With Alexandre Zhou, we obtain existence in the special case of the LSV model called regime switching local volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level.

10h10 – 10h50 : M. Tomasevic

“A new probabilistic interpretation of Keller-Segel model forchemotaxis, application to 1-d”

The Keller Segel (KS) model for chemotaxis is a two-dimensional system of parabolic or elliptic PDEs. Motivated by the study of the fully parabolic model using probabilistic methods, we give rise to a non linear SDE of McKean-Vlasov type with a highly non standard and singular interaction.
In this talk I will briefly introduce the KS model, point out some of the PDE analysis results on it and then, in detail, analyze our probabilistic interpretation in the case d=1.
This is a joint work with D. Talay and J-F. Jabir.

11h00 – 11h40  : M. Hauray

“Propagation du Chaos pour l’équation de Landau 3D homogène avec potentiels modérément mous”

11h40 – 12h20  : R. Cortez Milan

“Uniform propagation of chaos for the spatially homogeneous Boltzmann equation”

The spatially homogeneous Boltzmann equation models the evolution of the velocity distribution of a huge number of particles in a gas, subjected to elastic random binary collisions. In this work we study their corresponding finite stochastic $N$-particle system, and we are interested in the propagation of chaos property: the convergence, as $N\to\infty$ and for each time $t\geq 0$, of the empirical measure of the system towards the solution of the Boltzmann equation. Using recent probabilistic coupling techniques we find, under suitable moments assumptions on the initial distribution, an explicit uniform-in-time propagation of chaos rate of order almost $N^{-1/3}$ in squared $2$-Wasserstein distance for the Boltzmann equation in the Maxwell molecules case.

14h00 – 14h40  : M. Bossy

“Particle algorithm for McKean SDE: rate of convergence results when the drift kernel is non smooth”

14h40 – 15h20   : L. Szpruch

TBA

15h20 – 16h00  : S. Menozzi

TBA