On free boundary problems for the Atlas model

Abstract

For $n\in\mathbb{N}$, let $\{X^n_i\}$ be an infinite collection of Brownian particles on the real line where the leftmost particle $\min_iX^n_i(t)$ is given a drift $n$, and let $\mu^n_t=n^{-1}\sum_i\delta_{X^n_i(t)}$, $t\ge0$ denote the normalized configuration measure. The case where the initial particle positions follow a Poisson point process on $[0,\infty)$ of intensity $n\lambda$, $\lambda>0$ was studied where it was shown that $\mu^n_t$ converge, as $n\to\infty$, to a limit characterized by a Stefan problem of melting solid (respectively, freezing supercooled liquid) type when $\lambda\ge 2$ (respectively, $0<\lambda<2$). In this paper it is assumed that $\mu^n_0\to\mu_0$ in probability, where $\mu_0$ is supported on $[0,\infty)$ and satisfies a polynomial growth condition. Because $(y-x)^{-1}\mu_0((x,y])$, $0<x<y$ need not be bounded below or above by $2$, the model does not give rise to a Stefan problem of either of the above types. Under mild assumptions, it is shown that $\mu^n_t$ converge to a limit characterized by a free boundary problem involving measures. Under the additional assumption that $\mu_0(dx)\ge\lambda_0\,{\rm leb}_{[0,\infty)}(dx)$ for some $\lambda_0>0$, the free boundary exists as a continuous trajectory, and the process determined by the leftmost particle converges to it.