We consider the continuous resonant (CR) system of the 2-dimensional cubic
nonlinear Schrödinger (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.
Keywords
nonlinear Schrödinger equation, random data, Gibbs measure,
white noise measure, weak solutions, global solutions