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Ergodicity for the stochastic complex Ginzburg–Landau equations. (English) Zbl 1104.35078

The paper deals with the stochastic complex Ginzburg-Landau equation driven by a smooth noise in space. The aim of the paper is to study ergodicity for this equation under very general assumptions. The method used in this paper is a combination of two main ideas: a coupling method in a sufficiently general framework and the Foias-Prodi estimates. The author proves exponential convergence of the Markov transition semigroup toward a unique invariant probability measure. Two simple examples are discussed in the paper, which highlight the most important arguments in the method.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
37H99 Random dynamical systems
37L55 Infinite-dimensional random dynamical systems; stochastic equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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