Ergodic Dynamics of the Stochastic Swift-Hohenberg System

The Swift-Hohenberg fluid convection system with both local and nonlocal nonlinearities under the influence of white noise is studied. The objective is to understand the difference in the dynamical behavior in both local and nonlocal cases. It is proved that when sufficiently many of its Fourier modes are forced, the system has a unique invariant measure, or equivalently, the dynamics is ergodic. Moreover, it is found that the number of modes to be stochastically excited for ensuring the ergodicity in the local Swift-Hohenberg system depends {\em only} on the Rayleigh number (i.e., it does not even depend on the random term itself), while this number for the nonlocal Swift-Hohenberg system relies additionally on the bound of the kernel in the nonlocal interaction (integral) term, and on the random term itself