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.


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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