The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations

Abstract: This article deals with the approximation of a stochastic partial differential equation (SPDE) via amplitude equations. We consider an SPDE with a cubic nonlinearity perturbed by a general multiplicative noise that preserves the constant trivial solution and we study the dynamics around it for the deterministic equation being close to a bifurcation. Based on the separation of time-scales close to a change of stability, we rigorously derive an amplitude equation describing the dynamics of the bifurcating pattern. This allows us to approximate the original infinite dimensional dynamics by a simpler effective dynamics associated with the solution of the amplitude equation. To illustrate the abstract result we apply it to a simple one-dimensional stochastic Ginzburg-Landau equation.