Dottorato in MODELLI MATEMATICI PER L'INGEGNERIA, ELETTROMAGNETISMO E
NANOSCIENZE

Titolo della tesi: Mean Field Games for finite mixture models and Kolmogorov equation

This dissertation is concerned with Mean Field Games (MFG in short), a theory which aims to model the behaviour of an infinite number of rational agents who seek to minimize a common functional cost. The first part is focused on the applications of Mean Field Games to cluster analysis by means of finite mixture models, an important tool in the statistical analysis of data. Differently from the classical procedure where the optimal parameters of a mixture model are computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm, here an alternative approach based on Mean Field Games is proposed. Indeed, a multi-population Mean Field Games system is presented both in the case of Gaussian mixture and Bernoulli mixture model and then is generalized to mixture model of categorical distributions. Theoretical aspects are discussed and applications to some standard examples in cluster analysis are shown. The second part is concerned with the study of the hypoelliptic Hamilton Jacobi equation that naturally appears in Mean Field Games problems when the agents move according to a Langevin-type dynamics. The attention is given to the local existence of mild solution in the case of a polynomial growth of the gradient (only in the velocity variable). This exploits the decay estimates of the Kolmogorov semigroup. Moreover, the regularity is obtained for weak solutions using a duality technique.