Weak Solutions to Fokker–Planck Equations and Mean Field Games

We deal with systems of PDEs, arising in mean field games theory, where viscous Hamilton–Jacobi and Fokker–Planck equations are coupled in a forward- backward structure. We consider the case of local coupling, when the running cost depends on the pointwise value of the distribution density of the agents, in which case the smoothness of solutions is mostly unknown. We develop a complete weak theory, proving that those systems are well-posed in the class of weak solutions for monotone couplings under general growth conditions, and for superlinear convex Hamiltonians. As a key tool, we prove new results for Fokker–Planck equations under minimal assumptions on the drift, through a characterization of weak and renormalized solutions. The results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned.