Chaoticity of the stationary distribution of rank-based interacting diffusions

Julien Reygner (1, 2, 3)

1 Phys-ENS - Laboratoire de Physique de l'ENS Lyon
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

The mean-field limit of systems of rank-based interacting diffusions is known to be described by a nonlinear diffusion process. We obtain a similar description at the level of stationary distributions. Our proof is based on explicit expressions for the Laplace transforms of these stationary distributions and yields convergence of the marginal distributions in Wasserstein distances of all orders. We highlight the consequences of this result on the study of rank-based models of equity markets, such as the Atlas model.

Keywords: nonlinear diffusion process; stationary distribution; Rank-based interacting diffusions; chaoticity; Wasserstein distance

Domains: Probability [math.PR]

Julien Reygner. Chaoticity of the stationary distribution of rank-based interacting diffusions. Electronic Communications in Probability, 2015, 20 (60), pp.1-20. ⟨10.1214/ECP.v20-4063⟩. ⟨hal-01056364v2⟩