  We consider the continuous resonant (CR) system of the 2D cubic nonlinear
Schr{\"o}dinger (NLS) equation. This system arises in numerous instances as an
effective equation for the long-time dynamics of NLS in confined regimes (e.g.
on a compact domain or with a trapping potential). The system was derived and
studied from a deterministic viewpoint in several earlier works, which
uncovered many of its striking properties. This manuscript is devoted to a
probabilistic study of this system. Most notably, we construct global solutions
in negative Sobolev spaces, which leave Gibbs and white noise measures
invariant. Invariance of white noise measure seems particularly interesting in
view of the absence of similar results for NLS.
