  The stochastic Gross-Pitaevskii equation is used as a model to describe
Bose-Einstein condensation at positive temperature. The equation is a complex
Ginzburg Landau equation with a trapping potential and an additive space-time
white noise. Two important questions for this system are the global existence
of solutions in the support of the Gibbs measure, and the convergence of those
solutions to the equilibrium for large time. In this paper, we give a proof of
these two results in one space dimension. In order to prove the convergence to
equilibrium, we use the associated purely dissipative equation as an auxiliary
equation, for which the convergence may be obtained using standard techniques.
Global existence is obtained for all initial data, and not almost surely with
respect to the invariant measure.
