Abstract
We study a stochastic Complex Ginzburg–Landau (CGL) equation driven by a smooth noise in space and we establish exponential convergence of the Markov transition semi-group toward a unique invariant probability measure. Since Doob Theorem does not seem not to be useful in our situation, a coupling method is used. In order to make this method easier to understand, we first focus on two simple examples which contain most of the arguments and the essential difficulties.
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