CARDALIAGUET Pierre

We introduce a notion of weak solution of the master equation without idiosyncratic noise in Mean Field Game theory and establish its existence, uniqueness up to a constant and consistency with classical solutions when it is smooth. We work in a monotone setting and rely on Lions' Hilbert space approach. For the first-order master equation without idiosyncratic noise, we also give an equivalent definition in the space of measures and establish the well-posedness.

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In this thesis, we are mainly interested in three research topics, relatively independent, but nevertheless related through the thread of interactions and incentives, as highlighted in the introduction constituting the first chapter.In the first part, we present extensions of contract theory, allowing in particular to consider a multitude of players in principal-agent models, with drift and volatility control, in the presence of moral hazard. In particular, Chapter 2 presents a continuous-time optimal incentive problem within a hierarchy, inspired by the one-period model of Sung (2015) and enlightening in two respects: on the one hand, it presents a framework where volatility control occurs in a perfectly natural way, and, on the other hand, it highlights the importance of considering continuous-time models. In this sense, this example motivates the comprehensive and general study of hierarchical models carried out in the third chapter, which goes hand in hand with the recent theory of second-order stochastic differential equations (2EDSR). Finally, in Chapter 4, we propose an extension of the principal-agent model developed by Aïd, Possamaï, and Touzi (2019) to a continuum of agents, whose performances are in particular impacted by a common hazard. In particular, these studies guide us towards a generalization of the so-called revealing contracts, initially proposed by Cvitanić, Possamaï and Touzi (2018) in a single-agent model.In the second part, we present two applications of principal-agent problems to the energy domain. The first one, developed in Chapter 5, uses the formalism and theoretical results introduced in the previous chapter to improve electricity demand response programs, already considered by Aïd, Possamaï and Touzi (2019). Indeed, by taking into account the infinite number of consumers that a producer has to supply with electricity, it is possible to use this additional information to build the optimal incentives, in particular to better manage the residual risk implied by weather hazards. In a second step, chapter 6 proposes, through a principal-agent model with adverse selection, an insurance that could prevent some forms of precariousness, in particular fuel precariousness.Finally, we end this thesis by studying in the last part a second field of application, that of epidemiology, and more precisely the control of the diffusion of a contagious disease within a population. In chapter 7, we first consider the point of view of individuals, through a mean-field game: each individual can choose his rate of interaction with others, reconciling on the one hand his need for social interactions and on the other hand his fear of being contaminated in turn, and of contributing to the wider diffusion of the disease. We prove the existence of a Nash equilibrium between individuals, and exhibit it numerically. In the last chapter, we take the point of view of the government, wishing to incite the population, now represented as a whole, to decrease its interactions in order to contain the epidemic. We show that the implementation of sanctions in case of non-compliance with containment can be effective, but that, for a total control of the epidemic, it is necessary to develop a conscientious screening policy, accompanied by a scrupulous isolation of the individuals tested positive.

This thesis deals with the theory of mean-field games. Most of it is devoted to mean-field games in which players can interact through their state and control law. We will use the terminology mean-field control game to designate such games. First, we make a structural assumption, which essentially consists in saying that the optimal dynamics depends on the control law in a lipschitzian way with a constant less than one. In this case, we prove several existence results for solutions to the mean control field game system, and a uniqueness result in short time. In a second step, we set up a numerical scheme and perform simulations for population motion models. In a third step, we show the existence and uniqueness when the control interaction satisfies a monotonicity condition. The last chapter concerns a numerical solution algorithm for mean-field games of variational type and without interaction via the control law. We use a preconditioning strategy by a multigrid method to obtain a fast convergence.

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The aim of this paper is twofold. The first is to study the asymptotics of a parabolically scaled, continuous and space-time stationary in time version of the well-known Funaki-Spohn model in Statistical Physics. After a change of unknowns requiring the existence of a space-time stationary eternal solution of a stochastically perturbed heat equation, the problem transforms to the qualitative homogenization of a uniformly elliptic, space-time stationary, divergence form, nonlinear partial differential equation, the study of which is the second aim of the paper. An important step is the construction of correctors with the appropriate behavior at infinity.

This thesis is formulated in three parts with eight chapters and presents a research theme dealing with controlled processes/particles/interacting agents.In the first part of the thesis, we focus our attention on the study of interacting controlled processes representing a cooperative equilibrium, also known as Pareto equilibrium. A cooperative equilibrium can be seen as a situation where there is no way to improve the preference criterion of one agent without lowering the preference criterion of at least one other agent. It is now well known that this type of optimization problem is related, when the number of agents goes to infinity, to McKean-Vlasov optimal control. In the first three chapters of this thesis, we provide a precise mathematical answer to the link between these two optimization problems in different frameworks improving the existing literature, in particular by taking into account the control law while allowing a common noise situation.After studying the behavior of cooperative equilibria, we conclude the first part where we spend time in the analysis of the limit problem i.e. McKean-Vlasov optimal control, through the establishment of the dynamic programming principle (DPP) for this stochastic control problem.The second part of this thesis is devoted to the study of interacting controlled processes now representing a Nash equilibrium, also known as competitive equilibrium. A Nash equilibrium situation in a game is one in which no one has anything to gain by unilaterally leaving his own position. Since the pioneering work of Larsy - Lions and Huang - Malhamé - Caines, the behavior of Nash equilibria when the number of agents reaches infinity has been intensively studied and the associated limit game is known as Mean Field Games (MFG). In this second part, we first analyze the convergence of competitive equilibria to MFGs in a framework with the control law and with volatility control, then, the question of the existence of the MFG equilibrium in this context is studied.Finally, the last part, which consists of only one chapter, is devoted to some numerical methods for solving the limit problem i.e. McKean - Vlasov optimal control. Inspired by the proof of convergence of the cooperative equilibrium, we give a numerical algorithm to solve the McKean-Vlasov optimal control problem and prove its convergence. Then, we implement our algorithm using neural networks and test its efficiency on some application examples, namely mean-variance portfolio selection, the interbank systemic risk model and optimal liquidation with market impact.

We develop a splitting method to prove the well-posedness, in short time, of solutions for two master equations in mean field game (MFG) theory: the second order master equation, describing MFGs with a common noise, and the system of master equations associated with MFGs with a major player. Both problems are infinite dimensional equations stated in the space of probability measures. Our new approach simplifies, shortens and generalizes previous existence results for second order master equations and provides the first existence result for systems associated with MFG problems with a major player.

This volume provides an introduction to the theory of Mean Field Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a mean-field model for Nash equilibria in the strategic interaction of a large number of agents. Besides giving an accessible presentation of the main features of mean-field game theory, the volume offers an overview of recent developments which explore several important directions: from partial differential equations to stochastic analysis, from the calculus of variations to modeling and aspects related to numerical methods. Arising from the CIME Summer School "Mean Field Games" held in Cetraro in 2019, this book collects together lecture notes prepared by Y. Achdou (with M.

We obtain space-time Hölder regularity estimates for solutions of first-and second-order Hamilton-Jacobi equations perturbed with an additive stochastic forcing term. The bounds depend only on the growth of the Hamiltonian in the gradient and on the regularity of the stochastic coefficients, in a way that is invariant with respect to a hyperbolic scaling.

For two classes of Mean Field Game systems we study the convergence of solutions as the interest rate in the cost functional becomes very large, modeling agents caring only about a very short time-horizon, and the cost of the control becomes very cheap. The limit in both cases is a single first order integro-partial differential equation for the evolution of the mass density. The first model is a 2nd order MFG system with vanishing viscosity, and the limit is an aggregation equation. The result has an interpretation for models of collective animal behaviour and of crowd dynamics. The second class of problems are 1st order MFGs of acceleration and the limit is the kinetic equation associated to the Cucker-Smale model. The first problem is analyzed by PDE methods, whereas the second is studied by variational methods in the space of probability measures on trajectories.

We present an example of symmetric ergodic N´players differential games, played in memory strategies on the position of the players, for which the limit set, as N Ñ`8, of Nash equilibrium payoffs is large, although the game has a single mean field game equilibrium. This example is in sharp contrast with a result by Lacker [23] for finite horizon problems.

In this paper we investigate the well-posedness of backward or forward stochastic differential equations whose law is constrained to live in an a priori given (smooth enough) set and which is reflected along the corresponding "normal" vector. We also study the associated interacting particle system reflected in mean field and asymptotically described by such equations. The case of particles submitted to a common noise as well as the asymptotic system is studied in the forward case. Eventually, we connect the forward and backward stochastic differential equations with normal constraints in law with partial differential equations stated on the Wasserstein space and involving a Neumann condition in the forward case and an obstacle in the backward one.

We develop the counterpart of weak KAM theory for potential mean field games. This allows to describe the long time behavior of time-dependent potential mean field game systems. Our main result is the existence of a limit, as time tends to infinity, of the value function of an optimal control problem stated in the space of measures. In addition, we show a mean field limit for the ergodic constant associated with the corresponding Hamilton-Jacobi equation.

The goal of this paper is to study the long time behavior of solutions of the first-order mean field game (MFG) systems with a control on the acceleration. The main issue for this is the lack of small time controllability of the problem, which prevents to define the associated ergodic mean field game problem in the standard way. To overcome this issue, we first study the long-time average of optimal control problems with control on the acceleration: we prove that the time average of the value function converges to an ergodic constant and represent this ergodic constant as a minimum of a Lagrangian over a suitable class of closed probability measure. This characterization leads us to define the ergodic MFG problem as a fixed-point problem on the set of closed probability measures. Then we also show that this MFG ergodic problem has at least one solution, that the associated ergodic constant is unique under the standard mono-tonicity assumption and that the time-average of the value function of the time-dependent MFG problem with control of acceleration converges to this ergodic constant.

We consider Mean Field Games without idiosyncratic but with Brownian type common noise. We introduce a notion of solutions of the associated backward-forward system of stochastic partial differential equations. We show that the solution exists and is unique for monotone coupling functions. This the first general result for solutions of the Mean Field Games system with common and no idiosynctratic noise. We also use the solution to find approximate optimal strategies (Nash equilibria) for N-player differential games with common but no idiosyncratic noise. An important step in the analysis is the study of the well-posedness of a stochastic backward Hamilton-Jacobi equation.

This thesis deals with the study of the long time behavior of potential mean field games (MFG), independently of the convexity of the associated minimization problem. For the finite dimensional Hamiltonian system, similar problems have been treated by the weak KAM theory. We transpose many results of this theory to the context of potential mean field games. First, we characterize by ergodic approximation the boundary value associated with finite horizon MFG systems. We provide explicit examples in which this value is strictly greater than the energy level of the stationary solutions of the ergodic MFG system. This implies that the optimal trajectories of finite horizon MFG systems cannot converge to stationary configurations. Then, we prove the convergence of the minimization problem associated with finite horizon MFG to a solution of the critical Hamilton-Jacobi equation in the space of probability measures. Moreover, we show a mean field limit for the ergodic constant associated to the corresponding finite dimensional Hamilton-Jacobi equation. In the last part, we characterize the limit of the infinite horizon minimization problem that we used for the ergodic approximation in the first part of the manuscript.

Mean-field game systems (MFG) describe equilibrium configurations in differential games with an infinite number of infinitesimal agents. This thesis is structured around three different contributions to the theory of mean-field games. The main goal is to explore applications and extensions of this theory, and to propose new approaches and ideas to deal with the underlying mathematical issues. The first chapter first introduces the key concepts and ideas that we use throughout the thesis. We introduce the MFG problem and briefly explain the asymptotic connection with N-player differential games when N → ∞. We then present our main results and contributions. Chapter 2 explores an MFG model with a non-anticipatory interaction mode (myopic players). Unlike classical MFG models, we consider less rational agents who do not anticipate the evolution of the environment, but only observe the current state of the system, undergo changes, and take actions accordingly. We analyze the coupled PDE system resulting from this model, and establish the rigorous link with the corresponding N-Players game. We show that the population of agents can self-organize through a self-correcting process and converge exponentially fast to a well-known MFG equilibrium configuration. Chapters 3 and 4 concern the application of the MFG theory to the modeling of production and marketing processes of products with exhaustible resources (e.g. fossil fuels). In Chapter 3, we propose a variational approach for the study of the corresponding MFG system and analyze the deterministic limit (without demand fluctuations) in a regime where resources are renewable or abundant. In Chapter 4 we treat the MFG approximation by analyzing the asymptotic link between the N-player Cournot model and the MFG Cournot model when N is large. Finally, Chapter 5 considers an MFG model for the optimal execution of a portfolio of assets in a financial market. We explain our MFG model and analyze the resulting PDE system, then we propose a numerical method to compute the optimal execution strategy for an agent given its initial inventory, and present several simulations. Furthermore, we analyze the influence of trading activity on the intraday variation of the covariance matrix of asset returns. Next, we verify our findings and calibrate our model using historical trading data for a pool of 176 US stocks.

The Wasserstein space is the set of probability measures defined on a fixed domain and provided with the quadratic Wasserstein distance. In this work, we study variational problems in which the unknowns are Wasserstein-valued applications. When the starting space is a segment, i.e. when the unknowns are Wasserstein-valued curves, we are interested in models where, in addition to the action of the curves, terms penalizing congestion configurations are present. We develop techniques to extract regularity from the interaction between optimal density evolution (action minimization) and congestion penalization, and apply them to the study of mean-field games and the variational formulation of Euler equations.When the starting space is not only a segment but a domain of the Euclidean space, we consider only the Dirichlet problem, i.e. the minimization of the action (which can be called the Dirichlet energy) among all applications whose values on the edge of the starting domain are fixed. The solutions are called the harmonic applications with values in the Wasserstein space. We show that the different definitions of the Dirichlet energy present in the literature are in fact equivalent. that the Dirichlet problem is well posed under rather weak assumptions. that the superposition principle is defeated when the starting space is not a segment. that one can formulate a kind of maximum principle. and we propose a numerical method to compute these harmonic applications.

In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG " , we hence provide generic results to address these " MFG of controls " , before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.

Mean Field Game (MFG) systems describe equilibrium configurations in games with infinitely many interacting controllers. We are interested in the behavior of this system as the horizon becomes large, or as the discount factor tends to $0$. We show that, in the two cases, the asymptotic behavior of the Mean Field Game system is strongly related with the long time behavior of the so-called master equation and with the vanishing discount limit of the discounted master equation, respectively. Both equations are nonlinear transport equations in the space of measures. We prove the existence of a solution to an ergodic master equation, towards which the time-dependent master equation converges as the horizon becomes large, and towards which the discounted master equation converges as the discount factor tends to $0$. The whole analysis is based on the obtention of new estimates for the exponential rates of convergence of the time-dependent MFG system and the discounted MFG system.

In this paper we investigate the well-posedness of backward or forward stochastic differential equations whose law is constrained to live in an a priori given (smooth enough) set and which is reflected along the corresponding "normal" vector. We also study the associated interacting particle system reflected in mean field and asymptotically described by such equations. The case of particles submitted to a common noise as well as the asymptotic system is studied in the forward case. Eventually, we connect the forward and backward stochastic differential equations with normal constraints in law with partial differential equations stated on the Wasserstein space and involving a Neumann condition in the forward case and an obstacle in the backward one.

We present an example of symmetric ergodic N´players differential games, played in memory strategies on the position of the players, for which the limit set, as N Ñ`8, of Nash equilibrium payoffs is large, although the game has a single mean field game equilibrium. This example is in sharp contrast with a result by Lacker [23] for finite horizon problems.

Mean Field Game (MFG) systems describe equilibrium configurations in games with infinitely many interacting controllers. We are interested in the behavior of this system as the horizon becomes large, or as the discount factor tends to $0$. We show that, in the two cases, the asymptotic behavior of the Mean Field Game system is strongly related with the long time behavior of the so-called master equation and with the vanishing discount limit of the discounted master equation, respectively. Both equations are nonlinear transport equations in the space of measures. We prove the existence of a solution to an ergodic master equation, towards which the time-dependent master equation converges as the horizon becomes large, and towards which the discounted master equation converges as the discount factor tends to $0$. The whole analysis is based on the obtention of new estimates for the exponential rates of convergence of the time-dependent MFG system and the discounted MFG system.

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We develop the counterpart of weak KAM theory for potential mean field games. This allows to describe the long time behavior of time-dependent potential mean field game systems. Our main result is the existence of a limit, as time tends to infinity, of the value function of an optimal control problem stated in the space of measures. In addition, we show a mean field limit for the ergodic constant associated with the corresponding Hamilton-Jacobi equation.

The goal of this paper is to derive rigorously macroscopic traffic flow models from microscopic models. More precisely, for the microscopic models, we consider follow-the-leader type models with different types of drivers and vehicles which are distributed randomly on the road. After a rescaling, we show that the cumulative distribution function converge to the solution of a macroscopic model. We also make the link between this macroscopic model and the so-called LWR model.

Mean field games (MFG) are dynamic games with infinitely many infinitesimal agents. In this context, we study the efficiency of Nash MFG equilibria: Namely, we compare the social cost of a MFG equilibrium with the minimal cost a global planner can achieve. We find a structure condition on the game under which there exists efficient MFG equilibria and, in case this condition is not fulfilled, quantify how inefficient MFG equilibria are.

A large number of mathematical tools for modeling and analyzing multi-agent problems have recently been developed in the framework of optimal transport theory. In this thesis, we extend for the first time several of these concepts to problems from control theory. We prove several results on this subject, including necessary conditions for Pontryagin-like optimality in Wasserstein spaces, conditions ensuring the intrinsic regularity of optimal solutions, sufficient conditions for the emergence of different patterns, and an auxiliary result about the arrangements of certain singularities in sub-Riemannian geometry.

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For a mean field game model with a major and infinite minor players, we characterize a notion of Nash equilibrium via a system of so-called master equations, namely a system of nonlinear transport equations in the space of measures. Then, for games with a finite number N of minor players and a major player, we prove that the solution of the corresponding Nash system converges to the solution of the system of master equations as N tends to infinity.

This thesis introduces a new notion of solution for deterministic nonlinear evolution equations, called mild decoupled solutions. We revisit the links between Brownian Markovian backward differential equations (BMEEs) and semilinear parabolic PDEs by showing that, under very weak assumptions, BMEEs produce a unique mild decoupled solution of a PDE.We extend this result to many other deterministic equations such as Pseudo-EDPs, Integral Partial Derivative Equations (IPDEs), distributional drift PDEs, or trajectory dependent E(I)DPs. The solutions of these equations are represented via EDSRs which can be without reference martingale, or directed by cadlag martingales. In particular, this thesis solves the identification problem, which consists, in the classical case of a Markovian EDSR, in giving an analytical meaning to the process Z, second member of the solution (Y,Z) of the EDSR. In the literature, Y usually determines a viscosity solution of the deterministic equation and this identification problem is solved only when this viscosity solution has a minimum of regularity. Our method allows to solve this problem even in the general case of jumping EDSRs (not necessarily Markovian).

Mean Field Games with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. The aim of this paper is to provide a meaning of the PDE system associated with these games, the so-called Mean Field Game system with state constraints. For this, we show a global semiconvavity property of the value function associated with optimal control problems with state constraints.

This thesis deals with the study of new medium field game models. We first study optimal stopping and impulse control models in the absence of common noise. We construct for these models a notion of adapted solution for which we prove existence and uniqueness results under natural assumptions. Then, we focus on several properties of mean-field games. We study the limit of these models to pure evolution models when the players' anticipation tends to 0. We show the uniqueness of equilibria for strongly coupled systems (coupled by strategies) under certain assumptions. We then prove some regularity results on a master equation that models a mean field game with common noise in a discrete state space. We then present a generalization of the standard Uzawa algorithm and apply it to the numerical solution of some mean-field game models, in particular optimal stopping or impulse control. Finally, we present a concrete case of mean-field game that comes from problems involving a large number of connected devices in telecommunications.

Mean-field games (MFG) are a class of differential games in which each agent is infinitesimal and interacts with a huge population of agents. In this thesis, we raise the question of the actual formation of the MFG equilibrium. Indeed, since the game is very complex, it is unrealistic to assume that agents can actually compute the equilibrium configuration. This suggests that if the equilibrium configuration arises, it is because the agents have learned to play the game. Thus, the main question is to find learning procedures in mean-field games and to analyze their convergences to an equilibrium. We were inspired by learning schemes in static games and tried to apply them to our dynamic MFG model. We focus particularly on applications of fictitious play and online mirror descent on different types of mean field games: Potential, Monotonic or Discrete.

We introduce the notion of stable solution in mean field game theory: they are locally isolated solutions of the mean field game system. We prove that such solutions exist in potential mean field games and are local attractors for learning procedures.

This paper is concerned with the behavior of the ergodic constant associated with convex and superlinear Hamilton-Jacobi equation in a periodic environment which is perturbed either by medium with increasing period or by a random Bernoulli perturbation with small parameter. We find a first order Taylor's expansion for the ergodic constant which depends on the dimension d. When d = 1 the first order term is non trivial, while for all d ≥ 2 it is always 0. Although such questions have been looked at in the context of linear uniformly elliptic homogenization, our results are the first of this kind in nonlinear settings. Our arguments, which rely on viscosity solutions and the weak KAM theory, also raise several new and challenging questions.

This thesis is divided into three main parts, dealing with quasi-linear parabolic PDEs on a junction, stochastic diffusions on a junction, and optimal control also on a junction, with control at the junction point. We start in the first chapter by introducing a new class of non-degenerate quasi-linear PDEs, satisfying a non-linear and non-dynamic Neumann (or Kirchoff) condition at the junction point. We prove the existence of a classical solution, as well as its uniqueness. One of the motivations for studying this type of PDE is to make the link with the theory of optimal control on junctions, and to characterize the value function of this type of problem using the Hamilton Jacobi Bellman equations. Thus, in the next chapter, we formulate a proof giving the existence of a diffusion on a junction. This process admits a local time, whose existence and quadratic variation depend essentially on the assumption of ellipticity of the second order terms at the junction. We will formulate an Itô formula for this process. Thus, thanks to the results of these two Chapters, we will formulate in the last Chapter a stochastic control problem on junctions, with control at the junction point. The control space is that of probability measures solving a martingale problem. We prove the compactness of the space of admissible controls, as well as the principle of dynamic programming.

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In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG " , we hence provide generic results to address these " MFG of controls " , before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.

Mean Field Game systems describe equilibrium configurations in differential games with infinitely many infinitesimal interacting agents. We introduce a learning procedure (similar to the Fictitious Play) for these games and show its convergence when the Mean Field Game is potential.

This thesis contains two contributions to the space-time homogenization of the first order Hamilton-Jacobi equations. These equations are related to the modelling of road traffic. Finally, some results of homogenization in a nearly periodic environment are presented. The first chapter is devoted to the homogenization of an infinite system of coupled differential equations with delay time. This system is derived from a microscopic model of simple road traffic. The drivers follow each other on an infinite straight road and their reaction time is taken into account. The speed of each driver is assumed to be a function of the distance to the preceding driver: we speak of a "follow-the-leader" model. Thanks to a strict comparison principle, we show the convergence to a macroscopic model for reaction times lower than a critical value. In a second step, we show a counterexample to the homogenization for a reaction time higher than this critical value, for particular initial conditions. For this purpose, we perturb the stationary solution in which the vehicles are all equidistant at the initial times. The second chapter deals with the homogenization of a Hamilton-Jacobi equation whose Hamiltonian is spatially discontinuous. The associated traffic model is a straight road with an infinite number of traffic lights. The traffic lights are assumed to be identical, equally spaced and the time delay between two successive lights is assumed to be constant. We study the large-scale influence of this phasing on the traffic. We distinguish between the free road portion, which will be represented by a macroscopic model, and the traffic lights, which will be modeled by time-dependent flow limiters. The theoretical framework is the one by C. Imbert and R. Monneau (2017) for Hamilton-Jacobi equations on networks. The study consists in the theoretical homogenization, where the effective Hamiltonian depends on the phasing, and then the obtaining of qualitative properties of this Hamiltonian with the help of observations via numerical simulations. The third chapter presents results of homogenization in an almost periodic environment. First, we study an evolution problem with a stationary Hamiltonian, almost periodical in space. Using almost periodical arguments, we carry out in a second time a new proof of the homogenization result of the second chapter. The Hamiltonian is then periodic in time and almost periodic in space. We also have some open questions, especially in the case where the Hamiltonian is almost periodic in time-space, and in the case of a traffic model where the traffic lights are quite close, with therefore a microscopic model between the lights.

We prove, under some assumptions, the existence of correctors for the stochastic homoge-nization of of " viscous " possibly degenerate Hamilton-Jacobi equations in stationary ergodic media. The general claim is that, assuming knowledge of homogenization in probability, correctors exist for all extreme points of the convex hull of the sublevel sets of the effective Hamiltonian. Even when homogenization is not a priori known, the arguments imply existence of correctors and, hence, homogenization in some new settings. These include positively homogeneous Hamiltonians and, hence, geometric-type equations including motion by mean curvature, in radially symmetric environments and for all directions. Correctors also exist and, hence, homogenization holds for many directions for non convex Hamiltoni-ans and general stationary ergodic media.

We introduce the notion of stable solution in mean field game theory: they are locally isolated solutions of the mean field game system. We prove that such solutions exist in potential mean field games and are local attractors for learning procedures.

Mean Field Game systems describe equilibrium configurations in differential games with infinitely many infinitesimal interacting agents. We introduce a learning procedure (similar to the Fictitious Play) for these games and show its convergence when the Mean Field Game is potential.

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Mean-field game theory is a powerful formalism recently introduced to study stochastic optimization problems with a large number of agents. After recalling the basic principles of this theory and presenting some typical applications, we study in detail a stylized model of a seminar, of the mean field type. We derive an exact equation that allows us to predict the start time of the seminar and analyze different limit regimes, in which we arrive at approximate expressions of the solution. Thus we obtain a "phase diagram" of the problem. We then turn to a more complex population model with attractive group effects. Thanks to a formal analogy with the nonlinear Schrödinger equation, we show general evolution laws for the mean values of the problem, that the system verifies certain conservation laws and we develop approximations of variational type. This allows us to understand the qualitative behavior of the problem in the regime of strong interactions.

In this work, we study the homogenization of some front propagation problems in stationary and ergodic environments. In the first part, we study the stochastic homogenization of some non-local front propagation problems. In particular, a non-local version of the perturbed Evans test function method is given. The second part is devoted to the numerical approximation of the effective Hamiltonian which follows from the stochastic homogenization of the Hamilton-Jacobi equations. Error estimates between the numerical solutions and the effective Hamiltonian are established. In the third part, we are interested in the stochastic homogenization of problems of propagating fronts which evolve in the normal direction with a speed which can be unbounded. We show homogenization results in the case of i.i.d. media.

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This thesis deals with the modeling, analysis and numerical analysis of nonlinear and nonlocal partial differential equations with applications to road traffic. Road traffic can be modeled at different scales. In particular, one can consider the microscopic scale which describes the dynamics of each vehicle individually and the macroscopic scale which sees traffic as a fluid and describes traffic using macroscopic quantities such as vehicle density and average speed. In this thesis, using viscosity solution theory, we make the transition from microscopic to macroscopic models. The interest of this switch is that microscopic models are more intuitive and easy to handle to simulate particular situations (junctions, traffic lights,.) but they are not adapted to large simulations (to simulate the traffic in a whole city for example). On the contrary, macroscopic models are less easy to modify (to simulate a particular situation) but they can be used for large scale simulations. The idea is therefore to find the macroscopic model equivalent to a microscopic model that describes a specific scenario (a junction, a fork, different types of drivers, a school zone,.). The first part of this thesis contains a homogenization and numerical homogenization result for a microscopic model with different types of drivers. In a second part, homogenization and numerical homogenization results are obtained for microscopic models with a local disturbance (speed bump, school zone,.). Finally, we present a homogenization result in the context of a bifurcation.

In this paper we study Mean Field Game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We show that this price corresponds to the pressure field from the models of incompressible Euler's equations à la Brenier. By this observation we manage to obtain a minimal regularity, which allows to write optimality conditions at the level of single agent trajectories and to define a weak notion of Nash equilibrium for our model.

We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of Hamilton-Jacobi, ''viscous'' Hamilton-Jacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate ''periodizations'' of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization.

We provide Sobolev estimates for solutions of first order Hamilton-Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions are differentiable almost everywhere. The proof relies on an inverse H\"older inequality. Applications to mean field games are discussed.

We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation.

We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of Hamilton-Jacobi, ''viscous'' Hamilton-Jacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate ''periodizations'' of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization.

Existence and uniqueness of a weak solution for first order mean field game systems with local coupling are obtained by variational methods. This solution can be used to devise $\epsilon-$Nash equilibria for deterministic differential games with a finite (but large) number of players. For smooth data, the first component of the weak solution of the MFG system is proved to satisfy (in a viscosity sense) a time-space degenerate elliptic differential equation.

We consider a system of mean field games with local coupling in the deterministic limit. Under general structure conditions on the Hamiltonian and coupling, we prove existence and uniqueness of the weak solution, characterizing this solution as the minimizer of some optimal control of Hamilton-Jacobi and continuity equations. We also prove that this solution converges in the long time average to the solution of the associated ergodic problem.

We consider a system of mean field games with local coupling in the deterministic limit. Under general structure conditions on the Hamiltonian and coupling, we prove existence and uniqueness of the weak solution, characterizing this solution as the minimizer of some optimal control of Hamilton-Jacobi and continuity equations. We also prove that this solution converges in the long time average to the solution of the associated ergodic problem.

Existence and uniqueness of a weak solution for first order mean field game systems with local coupling are obtained by variational methods. This solution can be used to devise $\epsilon-$Nash equilibria for deterministic differential games with a finite (but large) number of players. For smooth data, the first component of the weak solution of the MFG system is proved to satisfy (in a viscosity sense) a time-space degenerate elliptic differential equation.

We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation.

We present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level-set convex Hamilton-Jacobi equations in i.i.d. random environments, the first quantitative homogenization results for these equations in the stochastic setting. By taking advantage of a connection between the metric approach to homogenization and the theory of first-passage percolation, we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime (away from flat spot of the effective Hamiltonian). In the sub-ballistic regime (on the flat spot), we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed, without further assumptions, at an arbitrarily slow rate. We identify a necessary and sufficient condition on the law of the Hamiltonian for an algebraic rate of convergence to hold in the sub-ballistic regime and show, under this hypothesis, that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization. Our methods are novel and quite different from the techniques employed in the periodic setting, although we benefit from previous works in both first-passage percolation and homogenization. The link between the rate of homogenization and the flat spot of the effective Hamiltonian, which is related to the nonexistence of correctors, is a purely random phenomenon observed here for the first time.

We investigate a two players zero sum differential game with incomplete information on the initial state: The first player has a private information on the initial state while the second player knows only a probability distribution on the initial state. This could be view as a generalization to differential games of the famous Aumann-Maschler framework for repeated games. In an article of the first author, the existence of the value in random strategies was obtained for a finite number of initial conditions (the probability distribution is a finite combination of Dirac measures). The main novelty of the present work consists in : first extending the result on the existence of a value in random strategies for infinite number of initial conditions and second - and mainly - proving the existence of a value in pure strategies when the initial probability distribution is regular enough (without atoms).

We consider degenerate second order mean field games systems with a local coupling. The starting point is the idea that mean field games systems can be understood as an optimality condition for optimal control of PDEs. Developing this strategy for the degenerate second order case, we discuss the existence and uniqueness of a weak solution as well as its stability (vanishing viscosity limit). Speaker: Daniela TONON.

We prove explicit estimates for the error in random homogenization of degenerate, second-order Hamilton-Jacobi equations, assuming the coefficients satisfy a finite range of dependence. In particular, we obtain an algebraic rate of convergence with overwhelming probability under certain structural conditions on the Hamiltonian.

We show that the long time average of solutions of first order mean field game systems in finite horizon is governed by an ergodic system of mean field game type. The well-posedness of this later system and the uniqueness of the ergodic constant rely on weak KAM theory.

We study a two-player, zero-sum, stochastic game with incomplete information on one side in which the players are allowed to play more and more frequently. The informed player observes the realization of a Markov chain on which the payoffs depend, while the non-informed player only observes his opponent's actions. We show the existence of a limit value as the time span between two consecutive stages vanishes. this value is characterized through an auxiliary optimization problem and as the solution of an Hamilton-Jacobi equation.

We study the long time average, as the time horizon tends to infinity, of the solution of a mean field game system with a nonlocal coupling. We show an exponential convergence to the solution of the associated stationary ergodic mean field game. Proofs rely on semiconcavity estimates and smoothing properties of the linearized system. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdA.

We introduce a new notion of pathwise strategies for stochastic differential games. This allows us to give a correct meaning to some statement asserted in [Cardaliaguet-Rainer 2009].

We study the homogenization of a $G$-equation which is advected by a divergence free stationary vector field in a general ergodic random environment. We prove that the averaged equation is an anisotropic deterministic G-equation and we give necessary and sufficient conditions in order to have enhancement. Since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions. In addition, as we show, the associated minimal (first passage) time function does not satisfy, in general, the uniform integrability condition which is necessary to apply the sub-additive ergodic theorem. We overcome these obstacles by (i) establishing a new reachability (controllability) estimate for the minimal function and (ii) constructing, for each direction and almost surely, a random sequence which has both a long time averaged limit (due to the sub-additive ergodic theorem) and stays (in the same sense) asymptotically close to the minimal time.

We study a two-player, zero-sum, stochastic game with incomplete information on one side in which the players are allowed to play more and more frequently. The informed player observes the realization of a Markov chain on which the payoffs depend, while the non-informed player only observes his opponent's actions. We show the existence of a limit value as the time span between two consecutive stages vanishes. this value is characterized through an auxiliary optimization problem and as the solution of an Hamilton-Jacobi equation.

In this paper, we study the characterization of geodesics for a class of distances between probability measures introduced by Dolbeault, Nazaret and Savar e. We first prove the existence of a potential function and then give necessary and suffi cient optimality conditions that take the form of a coupled system of PDEs somehow similar to the Mean-Field-Games system of Lasry and Lions. We also consider an equivalent formulation posed in a set of probability measures over curves.

The objective of this thesis is to study stochastic differential games with incomplete information. We consider a game with two opposing players who control a diffusion in order to minimize, respectively maximize a specific payoff. To model the incompleteness of information, we follow the famous approach of Aumann and Maschler. We assume that there are different states of nature in which the game can take place. Before the game starts, the state is chosen at random. In this thesis we establish a dual representation for differential stochastic games with incomplete information. Here, we make extensive use of the theory of stochastic backward differential equations (SDEs), which proves to be an indispensable tool in this study. Furthermore, we show how, under certain restrictions, this representation allows the construction of optimal strategies for the informed player. Then, using the dual representation, we give a particularly simple proof of the semiconvexity of the value function of differential games with incomplete information. Another part of the thesis is devoted to numerical schemes for stochastic differential games with incomplete information. In the last part we study optimal stopping games in continuous time, called Dynkin games, with incomplete information. We also establish a dual representation, which is used to determine optimal strategies for the informed player in this case.

This thesis studies different types of games with asymmetric information. For games with incomplete information on both sides, we present a discrete approximation of the value function based on the Mertens-Zamir operator. We also give an optimal strategy for the one-sided incomplete information game where players optimize a current payoff independent of the system state. In the framework of non-zero-sum games, we give the characterization of Nash equilibrium payments in mixed strategies, which is very similar to the "folk theorem" of repeated games. These equilibrium payments are in fact Nash equilibrium payments in publicly correlated strategies. Finally, we study a type of imperfectly observed game where one of the players is blind. We establish the existence of the value, characterized as the unique solution of a Hamilton-Jacobi equation in the Wasserstein space.

Our dissertation has three parts, all of which are related to the issue of lack of information in game theory. First, we study the notion of Blackwell's approximability in repeated games with vector payments by using techniques developed in the framework of qualitative differential games. Indeed, we reformulate the sufficient condition of approximability of a closed set (B-set) by the notion of a discriminative domain for an appropriate qualitative differential game. By introducing an auxiliary repeated game, we prove that a closed set is *-approachable (i.e., deterministically approachable) if and only if it contains a nonempty B-set. One of our main results is to establish the links between the behavioral strategies in repeated games and the non-anticipatory strategies in the approximability game. We also study a discounted infinite horizon differential game with lack of information on both sides. For this we follow the model introduced by Cardaliaguet and extend it to the infinite horizon framework. We first obtain a principle of dynamic subprogramming. Then we prove that the upper and lower value functions are respectively sub-solutions and over-solutions in the dual sense of the associated Hamilton Jacobi equation. Using a comparison principle we prove the uniqueness of the solution in the dual sense and thus the existence of the value. In the last chapter, we study a control system with probabilistic information about the initial state and extend the viability and invariance theorems to the Wasserstein space of probability measures. As an application we consider a Mayer-type optimal control problem where the state of the system is known according to a probability law. Following Frankowska's epigraphic approach we characterize the function as a unique proximal episolution of a Hamilton-Jacobi type equation.

This work deals with the study of front propagations governed by local and non-local laws. In the level line method, the front is seen as a 0-level line of an auxiliary function. The geometrical law of the front evolution corresponds to a Hamilton-Jacobi equation on this function, which we consider in the context of viscosity solutions. In non-local models, the main difficulty in proving existence or uniqueness results is the absence of inclusion principle between the fronts. In the level line method, this corresponds to an absence of comparison principle between functions, which makes the use of usual techniques impossible. The alternative use of fixed point methods associates to any non-local equation a family of local equations. Understanding the regularity of the solutions of the local equations, and in particular the perimeter of their level lines, then appears crucial in fixed point arguments. In chapter 1, we prove integral formulations of the local eikonal equation, from which we derive estimates on the perimeter of the level lines of its solutions. In the rest of the work, we are interested in non-local equations, and in particular in a notion of weak solution for these equations. Two non-local models, the dynamics of dislocations and a Fitzhugh-Nagumo system, are also studied in detail. In particular, results of existence, uniqueness and numerical approximation of weak solutions are given.

This thesis is devoted to a differential game - that is, a dual-control differential system in which one player seeks to move the state of the system into a given target while the other player seeks to keep the state of the system off the target. This is the target game. We study this game in the context of non-anticipatory strategies (by Elliot & Kalton) and feedback strategies (by Breakwell & Bernhard). For each strategy context, we define victory domains, which are the initial data sets from which a player can win regardless of the opponent's action. We show, in the context of non-anticipatory strategies, that the victory domains form a partition of the target's complement. We characterize the victory domains of each player using a set (the discriminant kernel) defined from geometric conditions inspired by viability theory (c. F. J. P. Aubin). Thanks to this characterization, we highlight a barrier property on the edge of the victory domains. We propose algorithms for computing victory domains that do not require the computation of trajectories.