Derivation of a Fractional Cross-Diffusion System as the Limit of a Stochastic Many-Particle System Driven by L\'{e}vy Noise

TL;DR

This paper rigorously derives a fractional cross-diffusion system as the limit of a stochastic multi-species particle system driven by Lévy noise, extending previous methods to fractional and non-local interactions.

Contribution

It introduces a new derivation of fractional cross-diffusion systems from stochastic particle models with Lévy noise, including well-posedness and non-negativity results.

Findings

Established convergence of regularized empirical measures to the macroscopic system

Proved well-posedness and non-negativity for the fractional cross-diffusion system

Extended classical techniques to fractional and Lévy-driven particle systems

Abstract

In this article a fractional cross-diffusion system is derived as the rigorous many-particle limit of a multi-species system of moderately interacting particles that is driven by L\'{e}vy noise. The form of the mutual interaction is motivated by the porous medium equation with fractional potential pressure. Our approach is based on the techniques developed by Oelschl\"ager (1989) and Stevens (2000), in the latter of which the convergence of a regularization of the empirical measure to the solution of a correspondingly regularized macroscopic system is shown. A well-posedness result and the non-negativity of solutions are proved for the regularized macroscopic system, which then yields the same results for the non-regularized fractional cross-diffusion system in the limit.