“Selection by vanishing common noise for potential finite state mean field games”
Martedì 29 Ottobre 2019, ore 14:30 - Aula-seminari 430 - Alekos Cecchin (Université de Nice Sophia-Antipolis, Francia)
By adding a suitable Wright-Fisher common noise in the dynamics of the law, uniqueness is restored for finite state mean field games, for any shape of the costs. We then focus on a class of mean field games in which the costs admit a potential in the simplex of probability measures. Firstly, by modifying the cost in the mean field game, we show that the unique solution of the randomized mean field game is equal to the unique minimizer of a suitable stochastic control problem stated in the simplex, called potential randomized game. Thus we study the limit as the intensity of the common noise vanishes. On the one hand, the optimal trajectories converge to the mean field game solutions which are minimizers of the corresponding potential game (not randomized), that is a deterministic optimal control problem in the simplex. On the other hand, the classical solution to the second order master equation converges to the gradient of the value function of the related potential game; we select a particular weak solution, in the sense of distributions, of the master equation, which can be written as multidimensional system of hyperbolic PDEs stated in the simplex. Lastly, we establish uniqueness of such solution in a suitable class.
Joint work with Francois Delarue.