Singular diffusion limit of a tagged particle in zero range processes with Sinai-type random environment
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917935277146112 |
|---|---|
| author | Hudiani, Marcel Landim, Claudio Sethuraman, Sunder |
| author_facet | Hudiani, Marcel Landim, Claudio Sethuraman, Sunder |
| contents | We derive a singular diffusion limit for the position of a tagged particle in zero range interacting particle processes on a one dimensional torus with a Sinai-type random environment via two steps. In the first step, a regularization is introduced by averaging the random environment over an $\varepsilon N$-neighborhood. With respect to such an environment, the microscopic drift of the tagged particle is in form $\frac{1}{N}W_\varepsilon'$, where $W_\varepsilon'$ is a regularized White noise. Scaling diffusively, we find the nonequilibrium limit of the tagged particle $x^\varepsilon_t$ is the unique weak solution of $d x_t^{\varepsilon} = 2\frac{Φ(ρ^{\varepsilon}(t, x_t^{\varepsilon}))}{ρ^{\varepsilon}(t, x_t^\varepsilon)} \,W_{\varepsilon}'(x_t^\varepsilon) + \sqrt{\frac{Φ(ρ^{\varepsilon}(t, x_t^\varepsilon))}{ρ^{\varepsilon}(t, x_t^\varepsilon)}} \,dB_t$, in terms of the hydrodynamic mass density $ρ^\varepsilon$ recently identified and homogenized interaction rate $Φ$.
In the second step, we show that $x^\varepsilon$, as $\varepsilon$ vanishes, converges in law to the diffusion $x^0$ described informally by $d x_t^0 = 2\frac{Φ(ρ^{0}(t, x_t^{0}))}{ρ^{0}(t, x_t^0)} \,W'(x_t^0) + \sqrt{\frac{Φ(ρ^{0}(t, x_t^0))}{ρ^{0}(t, x_t^0)}} \,dB_t$, where $W'$ is a spatial White noise and $ρ^0$ is the para-controlled limit of $ρ^\varepsilon$ also recently identified, solving the singular PDE $ \partial_t ρ^0 = \frac{1}{2}ΔΦ(ρ^0) - 2\nabla \big(W' Φ(ρ^0)\big)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular diffusion limit of a tagged particle in zero range processes with Sinai-type random environment Hudiani, Marcel Landim, Claudio Sethuraman, Sunder Probability Mathematical Physics 60K35, 60L40, 82C22, 82C44 We derive a singular diffusion limit for the position of a tagged particle in zero range interacting particle processes on a one dimensional torus with a Sinai-type random environment via two steps. In the first step, a regularization is introduced by averaging the random environment over an $\varepsilon N$-neighborhood. With respect to such an environment, the microscopic drift of the tagged particle is in form $\frac{1}{N}W_\varepsilon'$, where $W_\varepsilon'$ is a regularized White noise. Scaling diffusively, we find the nonequilibrium limit of the tagged particle $x^\varepsilon_t$ is the unique weak solution of $d x_t^{\varepsilon} = 2\frac{Φ(ρ^{\varepsilon}(t, x_t^{\varepsilon}))}{ρ^{\varepsilon}(t, x_t^\varepsilon)} \,W_{\varepsilon}'(x_t^\varepsilon) + \sqrt{\frac{Φ(ρ^{\varepsilon}(t, x_t^\varepsilon))}{ρ^{\varepsilon}(t, x_t^\varepsilon)}} \,dB_t$, in terms of the hydrodynamic mass density $ρ^\varepsilon$ recently identified and homogenized interaction rate $Φ$. In the second step, we show that $x^\varepsilon$, as $\varepsilon$ vanishes, converges in law to the diffusion $x^0$ described informally by $d x_t^0 = 2\frac{Φ(ρ^{0}(t, x_t^{0}))}{ρ^{0}(t, x_t^0)} \,W'(x_t^0) + \sqrt{\frac{Φ(ρ^{0}(t, x_t^0))}{ρ^{0}(t, x_t^0)}} \,dB_t$, where $W'$ is a spatial White noise and $ρ^0$ is the para-controlled limit of $ρ^\varepsilon$ also recently identified, solving the singular PDE $ \partial_t ρ^0 = \frac{1}{2}ΔΦ(ρ^0) - 2\nabla \big(W' Φ(ρ^0)\big)$. |
| title | Singular diffusion limit of a tagged particle in zero range processes with Sinai-type random environment |
| topic | Probability Mathematical Physics 60K35, 60L40, 82C22, 82C44 |
| url | https://arxiv.org/abs/2502.17365 |