On free boundary problems for the Atlas model

For $n\in \mathbb{N}$, let $\{X_i^n\}$ be an infinite collection of Brownian particles on the real line where the leftmost particle ${\min }_i{X}_i^n(t)$ is given a drift $n$, and let ${\mu }_t^n=n^{-1}{\sum }_i{\delta }_{X_i^n(t)}$, $t\ge 0$ denote the normalized configuration measure. The case where the initial particle positions follow a Poisson point process on $[0,\mathrm{\infty })$ of intensity $n\lambda$, $\lambda >0$ was studied where it was shown that ${\mu }_t^n$ converge, as $n\to \mathrm{\infty }$, 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 ${\mu }_0^n\to {\mu }_0$ in probability, where ${\mu }_0$ is supported on $[0,\mathrm{\infty })$ 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 ${\mu }_t^n$ converge to a limit characterized by a free boundary problem involving measures. Under the additional assumption that ${\mu }_0(dx)\ge {\lambda }_0{\mathrm{l}\mathrm{e}\mathrm{b}}_{[0,\mathrm{\infty })}(dx)$ for some ${\lambda }_0>0$, the free boundary exists as a continuous trajectory, and the process determined by the leftmost particle converges to it.