Topics in finite state mean field games and non-Markovian interacting spin systems

This Dissertation is devoted to the study of large stochastic systems of small interacting individuals and their macroscopic limit formulations, under symmetric properties of the interactions. The examples we consider belong to two separate contexts, depending on whether the individuals can control their dynamics or not. In the first case, treated in Part 1, we fall into the framework of N-player and mean field games, while in the latter, analyzed in Part 2, the resulting models are examples of interacting particle systems. More specifically, in the first part (Chapters 1-2) we focus on the convergence problem in mean field games, i.e. on the rigorous justification of mean field games as limits, when the number of players tends to infinity, of Nash equilibria of symmetric non-zero sum non-cooperative N-player games. In particular, we study finite state mean field games, where the state of each player belongs to a discrete finite space, analyzing separately the uniqueness case (Chapter 1) and a scenario with non-uniqueness of solutions to the mean field game (Chapter 2). In the second part of the Dissertation (Chapters 3-4) we study some examples of interacting spin systems, with non-Markovian individual dynamics, arising as proper modifications of classical ferromagnetic mean field spin systems dynamics. In particular, we focus on two mechanisms for relaxing the Markovianity: a state augmentation procedure, and the insertion of memory effects in the evolution. While one of the goals is still to rigorously justify the passage to a macroscopic description, the models of Part 2 present some features of independent interest, including phase transitions (Chapters 3-4), the emergence of self-sustained oscillations (Chapter 3), and the presence of multiple spatio-temporal scales phenomena (Chapter 4).