Adiabatic elimination and reduced probability distribution functions in spatially extended systems with a fluctuating control parameter

We obtain the stationary probability distribution functions of the order parameter near onset for the one-dimensional real Ginzburg-Landau and Swift-Hohenberg equations with a fluctuating control parameter. A perturbative expansion in the intensity of the fluctuations leads to a hierarchy of Fokker-Planck equations for conditional probability distribution functions that relate components of the order parameter that evolve in different time scales. Successive integration leads to a Fokker-Planck equation for the slowest mode, which we solve analytically for the models studied. In all cases, the probability distribution function above onset is of the form ${P(A}_0)\propto A_0^{\delta }{e}^{-\gamma A_0^2},$ where $A_0$ is the slow component of the order parameter and the values of δ and γ depend explicitly on the intensity of the fluctuations. Knowledge of ${P(A}_0)$ allows the calculation of an effective bifurcation threshold and of the moments of $A_0$ above threshold.