On the continuous resonant equation for NLS: II. Statistical study

TL;DR

This paper investigates the probabilistic properties of the continuous resonant system associated with 2D cubic NLS, demonstrating the existence of global solutions in negative Sobolev spaces and invariance of certain measures, including white noise.

Contribution

It provides the first probabilistic construction of global solutions for the CR system and shows measure invariance, extending understanding of long-time dynamics in low-regularity settings.

Findings

Constructed global solutions in negative Sobolev spaces.

Proved invariance of Gibbs and white noise measures.

White noise invariance is novel for NLS-related systems.

Abstract

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.