Jean-Pierre Eckmann, Martin Hairer
Uniqueness of the Invariant Measure for a 
Stochastic PDE Driven by Degenerate Noise
(439K, postscript)

ABSTRACT.  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.