Ricerc@Sapienza

We prove a representation formula of Hopf-Lax type for solutions to Hamilton-Jacobi equation involving a Caputo time-fractional derivative. Equations of this type are associated with optimal control problems where the controlled dynamics is given by a time-changed stochastic process describing the trajectory of a particle subject to random trapping effects.

We consider the variational structure of a time-fractional second-order mean field games (MFG) system. The MFG system consists of time-fractional Fokker–Planck and Hamilton–Jacobi–Bellman equations. In such a situation, the individual agent follows a non-Markovian dynamics given by a subdiffusion process. Hence, the results of this paper extend the theory of variational MFG to the subdiffusive situation.