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

TL;DR

This paper introduces a new discretization scheme for nonlinear Fokker-Planck-Kolmogorov equations that preserves key properties and converges to measure-valued solutions, with applications to Mean Field Games and pedestrian dynamics.

Contribution

It presents a novel discretization method that ensures non-negativity, mass conservation, and convergence to solutions, providing a new proof of existence for these equations.

Findings

Discretization scheme preserves non-negativity and mass.

Convergence to measure-valued solutions as parameters tend to zero.

Applications demonstrated in Mean Field Games and pedestrian models.

Abstract

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.