Rigorous mean-field limit and cross diffusion

The mean-field limit in a weakly interacting stochastic many-particle system for multiple population species in the whole space is proved. The limiting system consists of cross-diffusion equations, modeling the segregation of populations. The mean-field limit is performed in two steps: First, the many-particle system leads in the large population limit to an intermediate nonlocal diffusion system. The local cross-diffusion system is then obtained from the nonlocal system when the interaction potentials approach the Dirac delta distribution. The global existence of the limiting and the intermediate diffusion systems is shown for small initial data, and an error estimate is given.