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Communications in Mathematical Physics

, Volume 219, Issue 3, pp 523–565 | Cite as

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

  • J.-P. Eckmann
  • M. Hairer

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.

Keywords

Spatial Frequency Bounded Domain Invariant Measure High Spatial Frequency Landau Equation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • J.-P. Eckmann
    • 1
  • M. Hairer
    • 2
  1. 1.Département de Physique Théorique, Université de Genève, 1211 Genève, Switzerland.¶E-mail: Jean-Pierre.Eckmann@physics.unige.ch; Martin.Hairer@physics.unige.chCH
  2. 2.Section de Mathématiques, Université de Genève, 1211 Genève, SwitzerlandUS

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