Abstract. 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.

Abstract. We derive an amplitude equation for a stochastic partial differential equation (SPDE) of Swift-Hohenberg type with a nonlinearity that is composed of a stable cubic and an unstable quadratic term, under the assumption that the noise acts only on the constant mode. Due to the natural separation of timescales, solutions are approximated well by the slow modes. Nevertheless, via the nonlinearity, the noise gets transmitted to those modes too, such that multiplicative noise appears in the amplitude equation.

We consider a quite general class of SPDEs with quadratic and cubic nonlinearities and derive rigorously amplitude equations, using the natural separation of time-scales near a change of stability. We show that degenerate additive noise has the potential to stabilize or destabilize the dynamics of the dominant modes, due to additional deterministic terms arising in averaging.We focus on equations with quadratic and cubic nonlinearities and give applications to the Burgers' equation, the Ginzburg-Landau equation and generalized Swift-Hohenberg equation.