Validity of the stochastic Ginzburg-Landau approximation in higher space dimensions -A Wiener algebra approach-
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912849632165888 |
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| author | Logioti, Anna Schneider, Guido |
| author_facet | Logioti, Anna Schneider, Guido |
| contents | We consider an anisotropic $d$-dimensional Swift-Hohenberg model $ \mathcal{O}(\varepsilon^2) $-close to the first instability, where $ 0 < \varepsilon \ll 1 $ is a small perturbation parameter. This model for pattern formation is perturbed with additive noise in time and space. By a multiple scaling ansatz we derive a stochastic $ d $-dimensional Ginzburg-Landau equation for the approximate description of the bifurcating solutions. We prove the validity of the approximation by this amplitude equation on its natural time scale in case of $ d $-dimensional periodic domains of length $ \mathcal{O}(1/\varepsilon) $ for the Swift-Hohenberg model under suitable conditions on the additive noise. In detail, we prove the validity of this approximation for noise whose set of Fourier coefficients with respect to $ x $ is in $ \ell^1 $ for fixed $ t \geq 0 $. Moreover, we improve existing approximation results in the sense that the stable part of the noise can be larger. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18406 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Validity of the stochastic Ginzburg-Landau approximation in higher space dimensions -A Wiener algebra approach- Logioti, Anna Schneider, Guido Probability Analysis of PDEs We consider an anisotropic $d$-dimensional Swift-Hohenberg model $ \mathcal{O}(\varepsilon^2) $-close to the first instability, where $ 0 < \varepsilon \ll 1 $ is a small perturbation parameter. This model for pattern formation is perturbed with additive noise in time and space. By a multiple scaling ansatz we derive a stochastic $ d $-dimensional Ginzburg-Landau equation for the approximate description of the bifurcating solutions. We prove the validity of the approximation by this amplitude equation on its natural time scale in case of $ d $-dimensional periodic domains of length $ \mathcal{O}(1/\varepsilon) $ for the Swift-Hohenberg model under suitable conditions on the additive noise. In detail, we prove the validity of this approximation for noise whose set of Fourier coefficients with respect to $ x $ is in $ \ell^1 $ for fixed $ t \geq 0 $. Moreover, we improve existing approximation results in the sense that the stable part of the noise can be larger. |
| title | Validity of the stochastic Ginzburg-Landau approximation in higher space dimensions -A Wiener algebra approach- |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2601.18406 |