We propose to launch a research project on Hamilton-Jacobi (HJ) equations on networks, and more generally on heterogeneous structures. This theoretical problem has several potential applications, in particular to traffic flow theory, which is much studied from the point of view of conservation laws, but very little from the viewpoint of HJ equations. Whereas discrete control problems on networks have been much studied in the literature, there is almost nothing written on the case when the running cost and the dynamics vary with respect to time and state. The difficulty lies in the fact that, in a network, the set of admissible controls drastically changes from a point in the interior of an edge, where only one direction is admissible, to a vertex where the admissible directions are given by all the edges connected to it. Therefore, even if the data of the problem are regular, the corresponding Hamiltonian when restricted to the network has a discontinuous structure with respect to the spatial variable. This kind of discontinuity is at the cutting edge of the viscosity theory for HJ equations and most related problems are wide open. We think that there is room for a three years ANR-blanc project on the previously mentioned topic as well as on related problems about HJ equations. There will be four partners: Rennes (IRMAR), Paris 7 (LJJL), École Nationale des Ponts et Chaussées (CERMICS) and Tours (LMPT). We plan to investigate many aspects of the problems, both theoretical and practical. Trying to establish a robust theory of viscosity solutions for HJ equations on networks and heterogeneous structures will be a fundamental task. It will include the study of related and more general problems, such as HJ equations with space discontinuities. Another task will be to study the applications to traffic: we will start with models for traffic on a divergent junction, and gradually complexify the models to include micro-macro approaches, junctions with variable proportions of directions, convergent junctions and priority laws. We plan to compare our results with those obtained using conservation laws. In a third task, we plan to study singular perturbation problems for domains whose thickness tends to zero and see if at the limit, we recover the previously mentioned notion of viscosity solution on networks. We will also consider homogenization problems. In particular, we will study situations when a network becomes denser and denser, to model traffic in cities for example. We will also investigate differential games (leading to nonconvex Hamiltonians) and ask ourselves the same questions about the proper notion of viscosity solutions on networks. We will also consider mean field games on networks, with obvious applications in traffic theory, but also in many other situations. Finally, numerical methods will be proposed and analyzed. The general organization of the project will be classical for a project in mathematics: we will have frequent workshops of two days each in order to organize the research and exchange our ideas. We will also organize a mid-term school+workshop and a final conference in Tours and Rennes in order to invite international experts and to advert our results.

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.

We plan to study the mathematical aspects of percolation and first-passage percolation. We do not plan to focus on the critical two-dimensional percolation, which involves specific tools, such as the SLE process introduced by Schramm in 1999, and was for example the topic of ANR MAC2 (ANR-10-BLAN-0123). Suppose we immerse a large porous stone in a bucket of water. Consider the following closely related problems: - What is the probability that the center of the stone is wetted ? - What is the time needed for the center of the stone to be wetted ? Percolation is a model for the first problem. This is the study of connectedness properties of random graphs. Edges model pairs of close points of the stone between which the water can flow. It was introduced by Broadbent and Hammersley in 1957. First-passage percolation is a model for the second problem. This can be seen as the study of random metrics. The random distance between two points models the time needed for the water to flow from one point to the other. It can also be seen as a model of random growth or as a model of competition. It was introduced by Hammersley and Welsh in 1965. The models are closely related by their mere definition (first passage percolation is a refinement of percolation) but also because their study share many ideas and tools (coarse graining, coupling, FKG and BK inequalities, ...). The theory of these models is well developed, at least in the standard case (i.i.d. case on Z^d). However, there remains several intriguing and important questions and long-standing conjectures. Recent significant developments on these issues, some of which involving members of the team, give some real hope to make further significant progress on these problems. We see the conjectures as stimulations for our research. We aim to make some progress on these problems but, more generally, our aim is to investigate first-passage percolation, percolation and their links both in the standard case and in some less standard ones.

The Piecewise Deterministic Markov Processes (PDMP) are non-diffusive stochastic processes which naturally appear in many areas of applications as communication networks, neuron activities, biological populations or reliability of complex systems. Their mathematical study has been intensively carried out in the past two decades but many challenging problems remain completely open. This project aims at federating a group of experts with different backgrounds (probability, statistics, analysis, partial derivative equations, modelling) in order to pool everyone's knowledge and create new tools to study PDMPs. The main lines of the project relate to estimation, simulation and asymptotic behaviors (long time, large populations, multi-scale problems) in the various contexts of application.