  We consider the stochastic Ginzburg-Landau equation in a bounded domain. We
assume the stochastic forcing acts only on high spatial frequencies. The
low-lying frequencies are then only connected to this forcing through the
non-linear (cubic) term of the Ginzburg-Landau equation. Under these
assumptions, we show that the stochastic PDE has a unique invariant measure.
The techniques of proof combine a controllability argument for the low-lying
frequencies with an infinite dimensional version of the Malliavin calculus to
show positivity and regularity of the invariant measure. This then implies the
uniqueness of that measure.
