Validity of the stochastic Ginzburg-Landau approximation in higher space dimensions -A Wiener algebra approach-

TL;DR

This work rigorously justifies the stochastic Ginzburg-Landau amplitude equation as an accurate description for the bifurcating patterns of a $d$-dimensional anisotropic Swift-Hohenberg model with spatio-temporal noise, in large periodic domains of size $\mathcal{O}(1/\varepsilon)$. Using a Wiener-algebra framework, the authors decompose the noise into critical and stable parts, derive precise estimates for the stochastic terms through Ornstein-Uhlenbeck processes, and prove that the SH dynamics remain within $\mathcal{O}(\varepsilon^\beta)$ of the GL prediction on the slow time scale, with $\beta>1$ and high probability. The key contributions include improved noise-size bounds (stable part of order $\varepsilon^\beta$ rather than $\varepsilon^{5/2}$ in $L^2$), a detailed mode-filtering approach, and a rigorous multi-scale reduction in higher space dimensions. The results provide a robust, quantitative connection between stochastic pattern formation in SH and its GL amplitude description, enabling uncertainty quantification and extending rigorous justification to more complex, higher-dimensional settings.

Abstract

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.

Theorems & Definitions (15)

• Remark 2.1
• Remark 2.2
• Lemma 3.1
• Corollary 3.2
• Theorem 3.3
• Corollary 3.4
• Theorem 5.1
• Remark 5.2
• Remark 5.3
• Lemma 5.4