The Atlas Model and SDEs with Boundary Interaction
Philipp Jettkant
TL;DR
The article develops a probabilistic framework for the mean-field limit of the Atlas model, showing that the limiting dynamics are governed by a novel reflected McKean–Vlasov SDE with a moving boundary whose average local time grows at a constant rate. The boundary is represented by a random measure in general, with a continuous boundary recovered under regular initial profiles; the authors connect this to the supercooled Stefan problem and provide a thorough equivalence between hitting-time and local-time interaction forms. They establish tightness and convergence of the finite Atlas model to a generalized mean-field limit, prove pathwise uniqueness for generalized solutions, and develop existence results for both generalized and strong solutions via Schauder fixed-point and a superposition principle. The work offers a robust probabilistic counterpart to PDE approaches for Stefan-type moving boundary problems and advances the understanding of boundary-interacting mean-field dynamics in rank-based particle systems.
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.