On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications

In this work, we consider the discretization of some nonlinear Fokker-Planck-Kolmogorov equations. The scheme we propose preserves the nonnegativity of the solution, conserves the mass, and, as the discretization parameters tend to zero, has limit measure-valued trajectories which are shown to solve the equation. The main assumptions to obtain a convergence result are that the coefficients are continuous and satisfy a suitable linear growth property with respect to the space variable. In particular, we obtain a new proof of existence of solutions for such equations. We apply our results to some nonlinear examples, including Mean Field Games systems and variations of the Hughes model for pedestrian dynamics.