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

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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 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.

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Authors and Affiliations

  1. Département de Physique Théorique, Université de Genève, 1211 Genève, Switzerland.¶E-mail: Jean-Pierre.Eckmann@physics.unige.ch; Martin.Hairer@physics.unige.ch, , , , , , CH

    J.-P. Eckmann

  2. Section de Mathématiques, Université de Genève, 1211 Genève, Switzerland, , , , , , US

    M. Hairer

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  1. J.-P. Eckmann
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  2. M. Hairer
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Received: 10 September 2000 / Accepted: 13 December 2000

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Eckmann, JP., Hairer, M. Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise. Commun. Math. Phys. 219, 523–565 (2001). https://doi.org/10.1007/s002200100424

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  • DOI: https://doi.org/10.1007/s002200100424

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