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The purpose of this paper is to study the influence of large or unbounded domains on a stochastic PDE near a change of stability, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider the stochastic Swift--Hohenberg equation and derive rigorously the Ginzburg--Landau equation as a modulation equation for the amplitudes of the dominating modes. We verify that small global noise has the potential to stabilize the modulation equation, and thus to destroy the dominant pattern.

Modulation Equation for Stochastic Swift--Hohenberg Equation