From rank-based interacting particles to hyperbolic systems of conservation laws

Rank-based interacting particles are systems of diffusion processes evolving on the real line with drift and variance coefficients depending only on their rank. For a choice of coefficients describing mean-field interactions, these systems provide a probabilistic representation of the solution to scalar nonlinear parabolic evolution equations through propagation of chaos results. After describing this phenomenon, we shall use this representation to derive convergence to equilibrium of the solution to the evolution equation. Then, we shall address the small noise limit of rank-based interacting particles, in order to obtain an approximation of scalar hyperbolic conservation laws, and finally extend this procedure to hyperbolic systems of conservation laws.