On nonlinear systems of PDEs arising in the theory of large population differential games

This thesis is concerned with the study of stochastic differential games with many players, under structural hypotheses that differ from the classic ones of Mean Field Game theory. We focus on Nash equilibria and the systems of partial differential equations that describe them, within two main settings, namely games with sparse interactions and Generalised Mean Field Games. In the first part of the thesis, we deal with network games with interactions between players governed by sparse graphs. We introduce the concept of unimportance of distant players and provide two precise declinations of it, one for open-loop and one for closed-loop games. Related implications are also investigated. The main character of the second part is the Nash system, of parabolic equations of Hamilton-Jacobi-Bellman type, describing closed-loop equilibria. We make use of structural assumptions inspired by the unimportance of distant players to prove existence and uniqueness for a class of Nash systems in infinitely many dimensions. Afterwards, we enter the framework of Generalised Mean Field Games and, for some N-player nonsymmetric Nash systems under hypotheses of semimonotonicity, we prove certain a priori estimates historically known to be both hard to obtain and crucial for a rigorous derivation of the Master Equation directly from of the Nash system as N diverges. Making use of such estimates in this bottom-up approach to the large population limit of the Nash system, we conclude by proving that in our context suitable generalisations of both the Mean Field system and a weak form of the Master Equation can be obtained.