Amplitude equations for stochastic Swift-Hohenberg equatio

We consider a mathematical model for the Rayleigh-Benard convection, the stochastic Swift-Hohenberg equation: \[ \partial_t u = -(1+\partial_x^2)^2u+\varepsilon^2\nu u-u^3+\varepsilon^{\frac{3}{2}}\xi(t,x). \] 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.