From scattering to Fokker-Planck kinetic equations: the Glioblastoma multiforme as a prototype problem.

Kinetic equations provide a natural mathematical framework for bridging microscopic and macroscopic descriptions of complex systems. Originally developed in the context of gas dynamics and continuum mechanics, kinetic theory has been successfully extended to multi-agent systems arising in the life and social sciences.

Different microscopic descriptions naturally lead to different classes of kinetic equations: interaction rules give rise to Boltzmann collision-type kinetic equations, while Markov jump processes lead to scattering equations. Within this framework, the Fokker-Planck equation emerges as the quasi-invariant limit of collision-like equations, reminiscent of the grazing-collision limit in classical kinetic theory.

Taking glioblastoma, a highly aggressive brain tumor, as a motivating application, I will show how one can pass from a collision-based microscopic description to a jump-process formulation, and vice versa, in order to identify the most appropriate microscopic model for cell migration in this specific phenomenon. I will then present a rigorous derivation of Fokker--Planck equations from scattering equations, preserving not only the first moment but also the second moment of the probability distribution, which is the solution of the kinetic equation.

I will discuss results on existence, uniqueness, well-posedness, entropy decay, together with a structure-preserving numerical scheme for the Fokker-Planck equation. This framework also allows the computation of the mean first passage time, and estimates of the distance between probability measures corresponding to the different kinetic descriptions in terms of Fourier and Wasserstein metrics.