Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise
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 thelow-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.
KeywordsSpatial Frequency Bounded Domain Invariant Measure High Spatial Frequency Landau Equation
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