Ergodicity for the weakly damped stochastic non-linear Schr\"{o}dinger equations

We study a damped stochastic non-linear Schr\"{o}dinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markovian transition semi-group toward a unique invariant probability measure. This kind of method was originally developped to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schr\"{o}dinger equation in the one dimensional cubic case. We prove that the mixing property holds and that the rate of convergence to equilibrium is at least polynomial of any power.