A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations

TL;DR

This paper develops and analyzes a fully-discrete numerical scheme for systems of nonlinear, nonlocal Fokker-Planck-Kolmogorov equations, demonstrating its application to population dynamics and Mean Field Games.

Contribution

The authors extend a previous scheme for single FPK equations to systems, proving convergence and applying it to complex population and game models.

Findings

The scheme converges under certain conditions.

Effective in modeling interacting populations.

Applicable to Mean Field Game systems.

Abstract

We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied recently by the authors for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games.