Partial Differential Equations on Sets of Measures

ongoing

Partial differential equations (PDEs) are ubiquitous in science, whether it is in physics, engineering, biology or economics, as they naturally arise in the modelling of continuous objects. Recently, several PDEs have been derived to characterize complex objects such as: the best response by a player facing a continuum of adversarial players, the optimal control used to manage a distributed system or also the free energy or rate function of some distributed physical systems. All the associated PDEs are written on a set of measures on a smooth finite dimensional domain and raise new and difficult analytical challenges such as regularity and stability of the solutions, or the existence of weak notions of solutions. A systematic understanding of those equations is missing at the time, but it would lead to: derive rigorously the PDEs form the underlying models, justify numerical computations and, most importantly, to prove quantitative estimates and several properties of the modelled systems, which are otherwise out of reach because of the absence of explicit solutions. The key challenges targeted in this proposal are concentrated on two PDEs on spaces of measures, which are for the moment only understood in particular regimes, often very simplified compared to their original motivations. The first one is the mean field game (MFG) master equation: we intend to obtain a precise theory of regularity which will then help us both to extend the actual theory to more realistic MFGs and to have a better understanding of the stability of MFG equilibria. The second one is the Hamilton-Jacobi-Bellman equation: we aim to obtain much more general stability properties in order to treat practical mean field optimal control problems and mean field physical systems. Moreover, we also plan to introduce new PDEs modelling the optimal control of MFG master equations, thus raising new mathematical challenges. The design of numerical schemes for such equations will complement the program.