On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications
Provides a rigorous numerical framework for a class of PDEs relevant to mean-field games and crowd modeling, but the approach is incremental (discretization of known equations).
The authors propose a discretization scheme for nonlinear Fokker-Planck-Kolmogorov equations that preserves non-negativity and mass, and prove convergence to measure-valued solutions under mild assumptions. The method yields a new existence proof and is applied to Mean Field Games and pedestrian dynamics models.
In this work, we consider the discretization of some nonlinear Fokker-Planck-Kolmogorov equations. The scheme we propose preserves the non-negativity 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 several examples, including Mean Field Games systems and variations of the Hughes model for pedestrian dynamics.