Chaoticity of the stationary distribution of rank-based interacting diffusions

We consider Brownian diffusions on the real line, interacting through rank-dependent drifts. It is known that in the mean-field limit, such particle systems behave like independent copies of a so-called nonlinear diffusion process. We prove a similar asymptotic behaviour at the level of stationary distributions. Our proof is based on explicit expressions for the Laplace transforms of the stationary distributions of both the particle system and the nonlinear diffusion process, 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.