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

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

2000-09-15

arxiv.org/abs/nlin/0009028v1

Nonlinear Sciences

Chaotic Dynamics

45 pages, 0 figures, needs 3 latex runs

Scientific paper

10.1007/s002200100424

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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