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Ergodicity for a weakly damped stochastic nonlinear Schrödinger equation. (English) Zbl 1091.60010

The authors study a damped stochastic nonlinear Schrödinger equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, the authors establish convergence of the Markov transition semigroup toward a unique invariant probability measure. This kind of method was originally developed to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. The authors consider here a weakly dissipative equation, the damped nonlinear Schrödinger equation in the one-dimensional cubic case. The authors prove that the mixing property holds and that the rate of convergence to equilibrium is at least polynomial of any power.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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