La prochaine séance des Probabilités de Demain aura lieu **mercredi 30 novembre 2022 après-midi** à l’Institut Henri Poincaré (IHP, 11 Rue Pierre et Marie Curie, 75005).

**Programme de l'après-midi :**

- 14h-15h :
**Thierry Bodineau**:*Mean field vs. short range interactions*We will review some general results on the convergence of deterministic dynamics associated with mean field and short range interactions. Even though both types of microscopic dynamics share similarities, in particular the notion of propagation of chaos, we will see that the structure of their fluctuations is very different. This is related to the stability of the microscopic dynamics and to the dissipative properties of the corresponding limiting equations

- 15h-15h30 :
**Yoan Tardy**:*Collisions of the supercritical Keller-Segel particle system*We study a particle system naturally associated with the 2-dimensional Keller-Segel equation. It consists of N Brownian particles in the plane, interacting through a binary attraction in θ/(Nr), where r stands for the distance between two particles. When the intensity θ of this attraction is greater than 2, this particle system explodes in finite time. We assume that N>3θ and study what happens near an explosion in detail. There are two slightly different scenarios, depending on the values of N and θ, here is one: at the explosion, a cluster consisting of precisely k0 particles emerges, for some deterministic k0≥7 depending on N and θ. Just before the explosion, there are infinitely many (k0−1)-ary collisions. There are also infinitely many (k0−2)-ary collisions before each (k0−1)-ary collision. And there are infinitely many binary collisions before each (k0−2)-ary collision. Finally, collisions of subsets of 3,…,k0−3 particles never occur. The other scenario is similar except that there are no (k0−2)-ary collisions

- Pause
- 16h-16h30 :
**Samuel Daudin**:*Mean-field Control under Constraints in Law* - 16h30-17h :
**Louis-Pierre Chaintron**:*Gibbs principle on path space and link to stochastic control*

Gibbs principle identifies the probability distributions which minimize the relative entropy functional under constraints as Gibbs measures whose energy corresponds to the linearised constraints. We prove a general version of this principle for probability measures over the space of continuous paths when non-linear constraints are imposed at every time on marginal distributions. This kind of minimization problem arises when studying large deviations of i.i.d. particles under mean-field conditioning. In the case of stochastic diffusions, we show that optimizers are the solution of a quadratic stochastic control problem with constraints on the law of the process itself. This stochastic control problem has been studied by S. Daudin using more analytic tools, and its optimality conditions correspond to a forward-backward mean-field game (MFG) system. Using the Girsanov transform we eventually recover path measures from the MFG solutions. This work is a collaboration with Giovanni Conforti (CMAP, Ecole Polytechnique) and Julien Reygner (Cermics, ENPC).

We present a stochastic control problem where the probability distribution of the state is constrained to remain in some region of the Wasserstein space of probability measures. Re- formulating the problem as an optimal control problem for a (linear) Fokker-Planck equation, we derive optimality conditions in the form of a mean-field game system of partial differential equations. The effect of the constraint is captured by the presence, in this system, of a Lagrange multiplier which is a non-negative Radon measure over the time interval. Our main result is to exhibit geometric conditions on the constraint, under which this multiplier is bounded and optimal controls are Lipschitz continuous in time. For a second time, we explain how the stochastic control problem with constraints in law arises as the limit of control problems for large number of interacting agents subject to almost-sure constraints