- 00-361 Jean-Pierre Eckmann, Martin Hairer
- Uniqueness of the Invariant Measure for a
Stochastic PDE Driven by Degenerate Noise
Sep 15, 00
(auto. generated ps),
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Abstract. We consider the stochastic Ginzburg-Landau equation in a bounded
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.