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

We consider an anisotropic $d$-dimensional Swift-Hohenberg model $\mathcal{O}({\epsilon }^2)$-close to the first instability, where $0<\epsilon \ll 1$ 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 $\mathcal{O}(1\mathrm{/}\epsilon )$ 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 ${\mathrm{\ell }}^1$ for fixed $t\ge 0$. Moreover, we improve existing approximation results in the sense that the stable part of the noise can be larger.