Mean Field Games (MFG) is new and challenging mathematical topic which models the dynamics of a large number of interacting agents. It has many applications: economics, finance, social sciences, engineering,.. MFG are at the intersection of mean field theory, optimal control and stochastic analysis, calculus of variations, partial differential equations and scientific computing. Based on the internationally recognized expertise of its teams, the project intends to achieve major breakthroughs in 4 directions: the mean field analysis (i.e., the derivation of the macroscopic models from the microscopic ones); the mathematical analysis of news MFG models; their numerical analysis; the development of new applications. In this period of quick and worldwide expansion of the MFG modeling, it intends to foster the French leadership in the domain and to attract new French researchers coming from related areas.

Most physical, biological and economic phenomena can be described by evolution equations. Such equations can be linear or more often non-linear, reversible or non-reversible, deterministic or stochastic. It is interesting to know certain qualitative properties of their solutions, in particular their regularity and long time behavior. These properties can be studied by analytic, probabilistic or approximation methods. The reversible setting, in which the underlying operator of the partial differential equation is symmetric, or equivalently, in which the law of the stochastic process is unchanged by time reversal, is well understood. There the long time behavior and regularity properties are mainly studied by means of functional inequalities such as the Poincaré and logarithmic Sobolev inequalities. These inequalities give bounds on Liapunov functionals of the evolution (entropies); they can be obtained for instance by Bakry-Emery methods of using Liapunov conditions, which stem from MCMC methods. In certain models the dynamics can be better understood by introducing particle methods (hence, large systems of coupled differential equations) or, following F. Otto, by interpreting the evolution as a gradient flow of an entropy with respect to a Wasserstein distance, hence providing a natural time discretization. In the present proposal we wish to go beyond this academic setting and study the stability and stabilization for both more realistic and more complex evolution equations, and in particular non-reversible or/and degenerate : instances are kinetic Fokker-Planck type hypoelliptic equations, Navier-Stokes type equations, for polymers, mean field equations and systems of reaction-diffusion equations modeling chemical reactions. For such models the techniques used in the reversible setting must be refined and extended, but overall new analytic, probabilistic and numerical techniques must be developed : in particular we intend to stress on the numerical study of such models. The stability of numerical schemes is a fundamental issue whose study is based on discrete functional inequalities and on associated (jump) stochastic processes. Estimating the constants in these inequalities is a main question which we intend to study using tools such as the curvature of discrete spaces, extremal functions, or Liapunov functions as developed by members of the present proposal in other settings. The issues under consideration have analytic, probabilistic and numerical aspects : this is why the present proposal gathers researchers in the fields of PDEs, discrete and continuous Markov processes and numerical analysis.

During the past thirty years it has been realized that relativistic effects have a profound impact on chemistry. It may be asked whether relativity was the last train from physics to chemistry, or whether further refinement is needed by taking into account the effects of quantum electrodynamics (QED). The present proposal aims at providing a definite answer to this question as well as providing tools for its exploration. Previous studies indicate that the effect of QED on valence properties, such as electron affinities and ionization energies, is to reduce the relativistic effects by about five percent, which is rather modest. The situation for properties that depend on the electron density in the vicinity of nuclei is less clear. We therefore plan to investigate QED effects in chemistry with emphasis on such properties, notably on NMR parameters, on core and Mössbauer spectroscopies and on electric field gradients (which allows the determination of nuclear electric quadrupole moments). Another domain in which QED effects may come into play concerns spectroscopic tests of fundamental physics. Spectroscopic experiments of extreme precision have been carried out on atoms and molecules in order to probe the standard model of the universe, as well as alternative models. Examples of such experiments concern the non- conservation of parity of chiral molecules as well as the search for a possible electric dipole moment of fundamental particles such as the electron. These experiments depend on theory for guidance and for the extraction of the quantities of interest. Ultimately the combination of theory with atomic and molecular spectroscopy may allow the determination of physical observables normally obtained from high-energy experiments such as the large hadron collider, but this would require not only experiments, but also theoretical calculations of very high accuracy, hence the need to know the importance of QED effects for such properties. We address our objective in two steps. In a first pragmatic step we will incorporate effective QED potentials, already available for atoms, into molecular calculations. Our platform for development will be the DIRAC program which is presently the leading program for 2- and 4-component relativistic molecular calculations with extensive functionality for molecular properties. In a second, more ambitious and therefore more risk-prone step we plan to formulate and implement a variational approach to QED in the framework of existing quantum chemical methods such as Hartree-Fock and Density Functional Theory. A key challenge will be real-space renormalisation to curb the singularities known from previous QED work, but now to be carried out in the framework of finite basis set expansions. The success of this step hinges on the multidisciplinary character of the molQED team, involving theoretical chemists, physicists and mathematicians

The project is devoted to the study of Kinetic Mean Field Games (KMFGs), i.e. the limit of cooperative or non-cooperative games in very large populations, composed of interacting individuals, where each individual has a small influence on the global behaviour of the system and where the behaviour of the system depends on additional (internal) variables. Though the observables of KMFGs do not depend on the additional variables themselves (as they are the moments of the distribution function with respect to these additional variables), such internal variables have to be taken into account in the models and they have an influence in its behaviour. Moreover, since different time scales can be involved in the phenomenon, the kinetic approach allows to treat systems that cannot be modelled by purely macroscopic equations. The methods to be used come from the most recent developments in kinetic theory, as well as from the strategies developed in the last years for macroscopic mean field games. They include: existence theory of nontrivial remarkable states, linear and nonlinear asymptotic stability methods, entropy/entropy dissipation estimates, perturbative methods, renormalized solutions, etc. Our goal is, in particular, to extract information from systems which are far from known steady states. Moreover, through the identification of a scale parameter, we will study the limit, as the parameter tends to zero, of the system. Particular emphasis will be addressed to the numerical simulation of KMFG systems. As the unknown function depends - in principle - on several variables and as the KMFG is in principle both forward and backward in time, the numerical study of kinetic mean field game systems requires new approaches and massive parallelization techniques. The easy access to teraflop-based manycore computing (i.e. either 1-card GPU-like office computing, or clusters of manycores HPC) will allow to handle KMFG equations at the numerical level and produce accurate numerical schemes. Since the modelling is not completely stabilized in many of the problems under consideration, a collaboration with teams of economists or biologists is planned. The research project proposed here foresees the hiring of excellent postdoctoral researchers, coming from national or international high level universities and aims to give a substantial contribution in developing KMFG theory. Our development strategy consists in a fast dissemination of the scientific results that we will obtain. It will be done by producing publications in international reviewed journals and by means of communications in international conferences and workshops. Our works will provide contributions in applied mathematics journals but also, for example, in economics or biology journals. The interaction with industries and agencies is planned. Furthermore, we aim to widespread our results also in non-academic contexts, in order to show how applied mathematics can help the productive world. Finally, we shall create a web site where the pieces of information on the ANR project will be available to the scientific community.

This proposal lies at the interface between partial differential equations and probability theory. It aims at developping entropy methods and associated notions and techniques, and at applying them to various models in connection with several fields, such as physics (plasmas, Schrödinger, etc) and mathematical biology (notably the Patlak-Keller-Segel system). Special attention will be devoted to their interpretation and approximations in terms of particle systems. Such techniques are nowadays well understood in classical settings, and we intend to extend them to nonlinear and/or degenerate models, and to systems. The long time behaviour is of special interest to us, and for that we shall develop devoted and efficient tools, for instance in terms of functional inequalities. This proposal is in part intended for young researchers. It will focus on obtaining quantitative and optimal results as much as possible, in view of possible numerical applications that would allow interaction with researchers from other fields.