  We study the approach to equilibrium, described by a Gibbs measure, for a
system on a $d$-dimensional torus evolving according to a stochastic nonlinear
Schr\"odinger equation (SNLS) with a high frequency truncation. We prove
exponential approach to the truncated Gibbs measure both for the focusing and
defocusing cases when the dynamics is constrained via suitable boundary
conditions to regions of the Fourier space where the Hamiltonian is convex. Our
method is based on establishing a spectral gap for the non self-adjoint
Fokker-Planck operator governing the time evolution of the measure, which is
{\it uniform} in the frequency truncation $N$. The limit $N\to\infty$ is
discussed.
