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Uniqueness of the Invariant Measure for a Stochastic PDE Driven by Degenerate Noise

Jean-Pierre Eckmann·Martin Hairer
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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.

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Jean-Pierre Eckmann, Martin Hairer. 2000-09-15. Uniqueness of the Invariant Measure for a Stochastic PDE Driven by Degenerate Noise. https://doi.org/10.1007/s002200100424

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