Limit Invariant Measures for the Modified Stochastic Swift–Hohenberg Equation in a 3D Thin Domain

This work is concerned with the modified stochastic Swift–Hohenberg equation in a 3D thin domain. Although the diffusion motion of molecules is irregular with the interference of the film-fluid fluctuation, the invariant measure on the trajectory space reveals delicate transition of the dynamical behavior when the interior forces change. We therefore prove that the invariant measure of the system converges weakly to the unique counterpart of the stochastic Swift–Hohenberg equation in a 2D bounded domain with a concrete convergence rate, as the modified parameter and the thickness of the thin domain tend to zero. Furthermore, we address that the smooth density of the limit invariant measure fulfills a Fokker–Planck equation.