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}(\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.