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.

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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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