Convergence of Interaction Particle Systems toward Cross-Diffusion Equations

par Vincent Bansaye

jeudi 21 mai 2026, 14:00 → 15:00 Europe/Paris

435 (ENS de Lyon)

We consider spatially structured populations that interact locally. More precisely, two species move, reproduce, and die at rates that depend on the local density of the other species, that is, on the number of individuals of the other species present at the same site. Our goal is to compare the empirical measure describing the distribution of the two species with a deterministic cross-diffusion system (SKT-type PDEs), in the regime where both the number of sites and the local population sizes become large.
The main difficulty arises from the nonlinearity of the diffusion terms. Our approach relies on an intermediate comparison with a discrete or semi-discrete system (an ODE system in dimension 2M). The results are based on the development of quantitative stability estimates in this setting (duality lemmas), on the control of the associated martingales in Sobolev norms, and, to account for birth and death processes, on the derivation of large deviation estimates.

This talk is based on joint works with Ayman Moussa, Felipe Muñoz, and Alexandre Bertolino.