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.


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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