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 $μ^n_t=n^{-1}\sum_iδ_{X^n_i(t)}$, $t\ge0$ denote the normalized configuration measure. The case where the initial particle positions follow a Poisson point process on $[0,\infty)$ of intensity $nλ$, $λ>0$ was studied where it was shown that $μ^n_t$ converge, as $n\to\infty$, to a limit characterized by a Stefan problem of melting solid (respectively, freezing supercooled liquid) type when $λ\ge 2$ (respectively, $0<λ<2$). In this paper it is assumed that $μ^n_0\toμ_0$ in probability, where $μ_0$ is supported on $[0,\infty)$ and satisfies a polynomial growth condition. Because $(y-x)^{-1}μ_0((x,y])$, $0 0$, the free boundary exists as a continuous trajectory, and the process determined by the leftmost particle converges to it.