DELARUE , François

International audience

Cette note a pour but de donner un aperçu des travaux sur les équations aux dérivées partielles stochastiques singulières, qui ont valu la médaille Fields à Martin Hairer. Nous retraçons plus particulièrement le cheminement suivi par Martin Hairer pour aborder l'équation de Kardar-Parisi-Zhang et élaborer, à partir de là, la théorie plus...

We here gather in a single note several original probabilistic works devoted to the analysis of the C^(1,1) regularity of the solution to the possibly degenerate complex Monge-Ampère equation. The whole analysis relies on a probabilistic writing of the solution as the value function of a stochastic optimal control problem. Such a representation...

The purpose of this short article is to address a simple example of a game with a large number of players in mean field interaction when the graph connection between them is not complete but is of the Erdös-Renyi type. We study the quenched convergence of the equilibria towards the solution of a mean field game. To do so, we follow recent works...

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory....

No description

The purpose of this work is to provide an explicit construction of a strong Feller semigroup on the space of probability measures over the real line that additionally maps bounded measurable functions into Lipschitz continuous functions, with a Lipschitz constant that blows up in an integrable manner in small time. Our construction relies on a...

The zero-noise result for Peano phenomena of Bafico and Baldi (1982) is revisited. The original proof was based on explicit solutions to the elliptic equations for probabilities of exit times. The new proof given here is purely dynamical, based on a direct analysis of the SDE and the relative importance of noise and drift terms. The transition...

The goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential...

International audience

The paper is a continuation of the Kusuoka-Stroock programme of establishing smoothness properties of solutions of (possibly) degenerate partial differential equations by using probabilistic methods. We analyze here a class of semi-linear parabolic partial differential equations for which the linear part is a second order differential operator...

We here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d, the first one being forced by a non-degenerate random noise and the (n-1) other ones being degenerate. The system formed by the n equations satisfies a suitable Hörmander condition: the second equation feels the noise plugged...

This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions.Volume II tackles the analysis of mean field games in which...

The purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games for which the interaction between the players is of mean-field type in the sense that the private state on which a player bases his own strategy depends upon the empirical distribution of the private states of the other...

The goal of this paper is to demonstrate that common noise may serve as an exploration noise for learning the solution of a mean field game. This concept is here exemplified through a toy linear-quadratic model, for which a suitable form of common noise has already been proven to restore existence and uniqueness. We here go one step further and...