Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation

TL;DR

This paper investigates the long-term behavior of rank-based interacting diffusions through a probabilistic approach, establishing uniqueness, existence, and convergence to equilibrium of solutions to a scalar quasilinear parabolic equation.

Contribution

It introduces a probabilistic solution framework for the equation, proves uniqueness and existence via propagation of chaos, and analyzes long-time convergence to equilibrium.

Findings

Uniqueness of probabilistic solutions established.

Existence derived from propagation of chaos.

Solutions converge to equilibrium over time.

Abstract

We study a quasilinear parabolic Cauchy problem with a cumulative distribution function on the real line as an initial condition. We call 'probabilistic solution' a weak solution which remains a cumulative distribution function at all times. We prove the uniqueness of such a solution and we deduce the existence from a propagation of chaos result on a system of scalar diffusion processes, the interactions of which only depend on their ranking. We then investigate the long time behaviour of the solution. Using a probabilistic argument and under weak assumptions, we show that the flow of the Wasserstein distance between two solutions is contractive. Under more stringent conditions ensuring the regularity of the probabilistic solutions, we finally derive an explicit formula for the time derivative of the flow and we deduce the convergence of solutions to equilibrium.