Ricerc@Sapienza

We consider fractional models in order to describe phenomena where the associated dynamics well accords with so-called anomalous behaviours. We aim to model phenomena involving large number of observables where each observable is driven by a fractional diffusion and the number of particles increases according to a non-linear fractional growth. Fractional equations arise from applications of a large class of phenomena, namely population dynamics, cell growth, bacterial motion, smart materials, bird flight, pedestrian motion. The study of innovative materials such as polymers (it is well-known that textile) also brings about challenging mathematical questions concerning fractional calculus. Fractional diffusions are driven by governing equations with fractional operators in time. The symbol of the fractional operator has a Bernstein representation given in terms of Lévy measure. Thus, the theory of fractional calculus we consider here, well accords with the theory of the time changes for Markov processes. With respect to the dynamic of the population, we consider a number of observables where the underlying dynamic of the single observable is given by an associated time-changed process driven by the same fractional equation. Thus, observables or agents are supposed to be undistinguishable. We are interested in the application of mean field games to growth theory. Special attention will be payed for Pareto-like models and the probability distributions related to fractional equations. Moreover, we study dynamics associated with fractional epidemic models in the framework of mean field game. An example is given by the SIS model with states: Susceptible, Infected. Each agent can act on a control as vaccination, acceptance of social distancing, hand washing, et cetera. Our approach can be extended with many other possible states. In the asymptotic analysis we move from discrete state problems on continuous state problems.