Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.17669 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910384374415360 |
|---|---|
| author | Erhard, Dirk Hairer, Martin Xu, Tiecheng |
| author_facet | Erhard, Dirk Hairer, Martin Xu, Tiecheng |
| contents | We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=Δu(t,x) +ξ_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$. Here, the $ξ$-field is $\mathbb{R}$-valued, acting as a dynamic random environment, and $Δ$ represents the discrete Laplacian. We focus on the case where $ξ$ is given by a rescaled symmetric simple exclusion process which converges to an Ornstein--Uhlenbeck process. By scaling the Laplacian diffusively and considering the equation on a torus, we demonstrate that in dimension $d=2$, when a suitably renormalized version of the above equation is considered, the sequence of solutions converges in law. This resolves an open problem from~\cite{EH23}, where a similar result was shown in the three-dimensional case. The novel contribution in the present work is the establishment of a gradient bound on the transition probability of a fixed but arbitrary number of labelled exclusion particles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17669 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A scaling limit of the 2D parabolic Anderson model with exclusion interaction Erhard, Dirk Hairer, Martin Xu, Tiecheng Probability We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=Δu(t,x) +ξ_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$. Here, the $ξ$-field is $\mathbb{R}$-valued, acting as a dynamic random environment, and $Δ$ represents the discrete Laplacian. We focus on the case where $ξ$ is given by a rescaled symmetric simple exclusion process which converges to an Ornstein--Uhlenbeck process. By scaling the Laplacian diffusively and considering the equation on a torus, we demonstrate that in dimension $d=2$, when a suitably renormalized version of the above equation is considered, the sequence of solutions converges in law. This resolves an open problem from~\cite{EH23}, where a similar result was shown in the three-dimensional case. The novel contribution in the present work is the establishment of a gradient bound on the transition probability of a fixed but arbitrary number of labelled exclusion particles. |
| title | A scaling limit of the 2D parabolic Anderson model with exclusion interaction |
| topic | Probability |
| url | https://arxiv.org/abs/2403.17669 |