Uniqueness of the Invariant Measurefor a Stochastic PDE Driven by Degenerate Noise
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.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- 2001
- DOI:
- 10.1007/s002200100424
- arXiv:
- arXiv:nlin/0009028
- Bibcode:
- 2001CMaPh.219..523E
- Keywords:
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- Nonlinear Sciences - Chaotic Dynamics;
- Mathematics - Probability
- E-Print:
- 45 pages, 0 figures, needs 3 latex runs