Amplitude equations for stochastic Swift-Hohenberg equatio

We consider a mathematical model for the Rayleigh-Benard convection, the stochastic Swift-Hohenberg equation: $${\partial }_{t}u=-(1+{\partial }_x^2{)}^{2}u+{\epsilon }^2\nu u-u^3+{\epsilon }^{\frac{3}{2}}\xi (t,x).$$ Near its change of stability, the fluid's motion can be described in a multiscale setting as the product of a slowly varying amplitude equation and a faster periodic wave. After an introduction to the problem in its deterministic setting, we'll review some known stochastic results and see some recent developments in the unbounded space domain setting.