Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.09811 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913649857134592 |
|---|---|
| author | Gerolla, Luca Hairer, Martin Li, Xue-Mei |
| author_facet | Gerolla, Luca Hairer, Martin Li, Xue-Mei |
| contents | We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions $d \geq 3$ with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and coloured in space with non-integrable spatial covariance that decays at the rate of $|x|^{-κ}$ at infinity, where $κ\in (2, d)$. Inspired by recent studies on SHE and KPZ equations driven by noise with compactly supported spatial correlation, we demonstrate that the correlations persist in the large-scale limit. The fluctuations of the diffusively scaled solution converge to the solution of a stochastic heat equation with additive noise whose correlation is the Riesz kernel of degree $-κ$. Moreover, the fluctuations converge as a distribution-valued process in the optimal Hölder topologies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_09811 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fluctuations of stochastic PDEs with long-range correlations Gerolla, Luca Hairer, Martin Li, Xue-Mei Probability Mathematical Physics 60H15, 60H05, 60F05 We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions $d \geq 3$ with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and coloured in space with non-integrable spatial covariance that decays at the rate of $|x|^{-κ}$ at infinity, where $κ\in (2, d)$. Inspired by recent studies on SHE and KPZ equations driven by noise with compactly supported spatial correlation, we demonstrate that the correlations persist in the large-scale limit. The fluctuations of the diffusively scaled solution converge to the solution of a stochastic heat equation with additive noise whose correlation is the Riesz kernel of degree $-κ$. Moreover, the fluctuations converge as a distribution-valued process in the optimal Hölder topologies. |
| title | Fluctuations of stochastic PDEs with long-range correlations |
| topic | Probability Mathematical Physics 60H15, 60H05, 60F05 |
| url | https://arxiv.org/abs/2303.09811 |