Large deviations for diffusions interacting through their ranks

TL;DR

This paper establishes a Large Deviations Principle for large systems of diffusions interacting through their ranks, providing explicit rate functions and linking the particle density to McKean-Vlasov and porous medium equations.

Contribution

It presents the first explicit LDP for rank-interacting diffusions with both drift and diffusion interactions, including new regularity results for related PDEs.

Findings

Explicit large deviations rate function derived

Unique solution of McKean-Vlasov equation identified

Porous medium equation describes the evolution of the distribution

Abstract

We prove a Large Deviations Principle (LDP) for systems of diffusions (particles) interacting through their ranks, when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the approriate McKean-Vlasov equation and that the corresponding cumulative distribution function evolves according to the porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of a LDP for interacting diffusions, where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for a certain tilted version of the porous medium equation.