The Atlas Model and SDEs with Boundary Interaction

TL;DR

This paper investigates the mean-field limit of the Atlas model, revealing a new class of SDEs with boundary interactions and local time dependence, connecting particle systems to measure-valued boundaries.

Contribution

It introduces a novel SDE with boundary reflection derived from the Atlas model, extending the understanding of mean-field limits with local time interactions.

Findings

Convergence of the particle system to the SDE with boundary interaction.

Boundary represented by a measure, with continuous boundary for regular initial profiles.

Reformulation via McKean--Vlasov SDEs with hitting and local times.

Abstract

We study the mean-field limit of the Atlas model and its connection to SDEs with dependence on the distribution of hitting and local times. The Atlas model describes a system of Brownian particles on the real line, where only the lowest ranked particle receives a positive drift, proportional to the number of particles. We show that in the mean-field limit the particle system converges to a novel SDE with reflection at a moving boundary, whose motion is such that the average local time spent at the boundary grows at a constant rate. In general, the boundary is represented by a measure, so the reflection must be interpreted in a relaxed sense. However, for sufficiently regular initial particle profiles, we prove that the boundary is a continuous function. Our analysis relies on a reformulation of the problem via McKean--Vlasov SDEs with interaction through hitting and local times.