Periodic space-time homogenisation of the $ϕ^4_2$ equation
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909409530085376 |
|---|---|
| author | Hairer, Martin Singh, Harprit |
| author_facet | Hairer, Martin Singh, Harprit |
| contents | We consider the homogenisation problem for the $ϕ^4_2$ equation on the torus $\mathbb{T}^2$, namely the behaviour as $\varepsilon \to 0$ of the solutions to the equation suggestively written as $$ \partial_t u_\varepsilon - \nabla\cdot {A}(x/\varepsilon,t/\varepsilon^2) \nabla u_\varepsilon = -u^3_\varepsilon +ξ$$ where $ξ$ denotes space-time white noise and $A: \mathbb{T}^2\times \mathbb{R}$ is uniformly elliptic, periodic and Hölder continuous. When the noise is regularised at scale $δ\ll 1$ we show that any joint limit $\varepsilon,δ\to 0$ recovers the classical dynamical $ϕ^4_2$ model. In certain regimes or if the regularisation is chosen in a specific way adapted to the problem, we show that the counterterms can be chosen as explicit local functions of $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15788 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Periodic space-time homogenisation of the $ϕ^4_2$ equation Hairer, Martin Singh, Harprit Analysis of PDEs Probability 60H17, 35B27 We consider the homogenisation problem for the $ϕ^4_2$ equation on the torus $\mathbb{T}^2$, namely the behaviour as $\varepsilon \to 0$ of the solutions to the equation suggestively written as $$ \partial_t u_\varepsilon - \nabla\cdot {A}(x/\varepsilon,t/\varepsilon^2) \nabla u_\varepsilon = -u^3_\varepsilon +ξ$$ where $ξ$ denotes space-time white noise and $A: \mathbb{T}^2\times \mathbb{R}$ is uniformly elliptic, periodic and Hölder continuous. When the noise is regularised at scale $δ\ll 1$ we show that any joint limit $\varepsilon,δ\to 0$ recovers the classical dynamical $ϕ^4_2$ model. In certain regimes or if the regularisation is chosen in a specific way adapted to the problem, we show that the counterterms can be chosen as explicit local functions of $A$. |
| title | Periodic space-time homogenisation of the $ϕ^4_2$ equation |
| topic | Analysis of PDEs Probability 60H17, 35B27 |
| url | https://arxiv.org/abs/2311.15788 |