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.
The project is characterized by a multidisciplinary point of view and profits of the synergy of the complementary expertise involved, namely: Probability and Statistic, for the definition of the important quantities to be considered; Mathematical Analysis and Applied Mathematics, to construct and investigate models; Numerical Analysis, to implement the required simulations. The purposes of the present project can be explained by the following main arguments which are included.
I) From Logistic equations to fractional logistic equations (and fractional systems).
This is a well-known Verhulst model a prototype for many other studies and also the basic equation for coupling and the study of more complicated processes. Its fractional logistic equation has been discussed in several context and it was not completely solved from two point of view. What does it means fractional setting of this equation since the equation is non-linear and Laplace transform method is not applicable. Then, it is not so direct to give fractional logistic equation with respect to this problem we give a study for short and large time. Recent papers of people of the research group give a partial answer to this question. Just to describe briefly we found the connection with Euler¿s numbers. This is an intriguing argument and leads to further directions of research. The potential field of applications are so many as evidenced by the numerous articles in the literature concerning the standard case.
II) From mean field games to fractional mean field games.
In the standard Mean Field Games model, the dynamics of the single agent is governed by a Gaussian diffusion process. Hence the underlying environment has no role in the problem or, in other words, is isotropic, an assumption not satisfied in several applications. On the contrary, subdiffusive processes display local motion occasionally interrupted by long sojourns, a trapping effects due to the anisotropy of the medium. An alternative viewpoint is given by the memory effect or the complex interactions between agents introduced by the (fractional or non-local) dependence structure. Aim of this part of the project is to introduce a class of MFG model in which the dynamics of the agent is subdiffusive rather diffusive as in the Lasry-Lions model. A subdiffusive regime is considered to be a better model not only for several transport phenomena in physics, but also, for example, in the study of volatility of financial markets, bacterial motion, bird flight, etc. According to the optimal control interpretation of the problem, we get a system involving fractional time derivatives for the HamiltonJacobi-Bellman and the Fokker-Planck equations. A first important point is to understand the correct formulation of the MFG system in this framework. Time-fractional FP equation governing the evolution of the PDF of a subdiffusive process were first derived for the case of a space-dependent drift and constant diffusion coefficient, hence the theory has been progressively generalized to include the case of space-time dependent coefficients which is relevant for our study.
III) From memory kernels for viscoelastic materials to memory kernels for smart materials.
In literature, classes of integro-differential equation representing evolution process of viscoelastic materials as oscillations are studied. From a mathematical point of view the study of innovative materials as polymers leads to the Burger model where the memory term is the sum of decreasing exponentials functions. This is a generalization of the well-known models as Kelvin-Voigt solid models (spring in parallel with a dashpot) and Maxwell fluid models (spring in series with a dashpot). An analysis may be done when we consider Mittag-Leffler functions. As well-known, roughly speaking, the Mittag-Leffler functions generalize the exponential functions and, in the linear case, the analysis will be done although it may contain unexpected difficulties. In particular, the study of oscillations leads to a wave-integro-differential equations. A spectral study is needed to understand the behaviour of the solution and it is preliminary to reachability problem in control theory. We have described the closer new results to our research interest. Our contributions are published in international Journals. Moreover, senior participants have been invited in national and international conferences to give lectures on related results.