Ergodicity for a weakly damped stochastic nonlinear Schrödinger equation (Q816564)

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.