On free boundary problems for the Atlas model
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908458819780608 |
|---|---|
| author | Atar, Rami Budhiraja, Amarjit |
| author_facet | Atar, Rami Budhiraja, Amarjit |
| contents | 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 $(y-x)^{-1}μ_0((x,y])$, $0<x<y$ 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 $μ^n_t$ converge to a limit characterized by a free boundary problem involving measures. Under the additional assumption that $μ_0(dx)\geλ_0\,{\rm leb}_{[0,\infty)}(dx)$ for some $λ_0>0$, the free boundary exists as a continuous trajectory, and the process determined by the leftmost particle converges to it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15479 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On free boundary problems for the Atlas model Atar, Rami Budhiraja, Amarjit Probability 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 $(y-x)^{-1}μ_0((x,y])$, $0<x<y$ 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 $μ^n_t$ converge to a limit characterized by a free boundary problem involving measures. Under the additional assumption that $μ_0(dx)\geλ_0\,{\rm leb}_{[0,\infty)}(dx)$ for some $λ_0>0$, the free boundary exists as a continuous trajectory, and the process determined by the leftmost particle converges to it. |
| title | On free boundary problems for the Atlas model |
| topic | Probability |
| url | https://arxiv.org/abs/2507.15479 |