Periodicity and longtime diffusion for mean field systems in $\mathbb{R}^d$

Eric Luçon (1, 2) , Christophe Poquet (3)

1 FP2M - Fédération Parisienne de Modélisation Mathématique
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
3 PSPM - Probabilités, statistique, physique mathématique

We study in this paper the longtime behavior of some large but finite populations of interacting stochastic differential equations whose (infinite population) limit Fokker-Planck PDE admits a stable periodic solution. We show that the empirical measure for the population of size $N$ stays close to the periodic solution, but with a random dephasing at the timescale $Nt$ that converges weakly to a Brownian motion with constant drift.

Domaines: Probabilités [math.PR]

Eric Luçon, Christophe Poquet. Periodicity and longtime diffusion for mean field systems in $\mathbb{R}^d$. 2021. ⟨hal-03280091⟩