标题： 关于Atlas模型的自由边界问题 显示英文标题

标题： On free boundary problems for the Atlas model

摘要： 对于$n\in\mathbb{N}$，令$\{X^n_i\}$是实线上无限的布朗粒子集合，其中最左边的粒子$\min_iX^n_i(t)$被赋予一个漂移$n$，并令$\mu^n_t=n^{-1}\sum_i\delta_{X^n_i(t)}$、$t\ge0$表示归一化的配置测度。 当初始粒子位置在$[0,\infty)$上服从强度为$n\lambda$,$\lambda>0$的泊松点过程时，该情况被研究，结果显示当$n\to\infty$时，$\mu^n_t$收敛到由熔化固体（分别，过冷液体冻结）类型的 Stefan 问题所表征的极限，当$\lambda\ge 2$（分别，$0<\lambda<2$）时。 在本文中假设 $\mu^n_0\to\mu_0$在概率上成立，其中$\mu_0$支持在$[0,\infty)$上并满足多项式增长条件。 由于$(y-x)^{-1}\mu_0((x,y])$， $0<x<y$不必由$2$从下方或上方有界，因此该模型不会产生上述类型的Stefan问题。 在适度的假设下，证明了$\mu^n_t$收敛到由涉及测度的自由边界问题所表征的极限。 在额外假设$\mu_0(dx)\ge\lambda_0\,{\rm leb}_{[0,\infty)}(dx)$对某个$\lambda_0>0$成立的情况下，自由边界作为连续轨迹存在，由最左粒子确定的过程收敛于它。 显示英文摘要

摘要： 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.