Revisiting Bourgain's probabilistic construction of solutions to the 2-$d$ cubic NLS

TL;DR

This paper revisits Bourgain's 1996 probabilistic method for constructing solutions to the 2D cubic NLS, streamlining the proof using recent advances in random tensor estimates.

Contribution

It provides a simplified and more efficient proof of Bourgain's invariance result by incorporating recent developments in random tensor analysis.

Findings

Streamlined proof of Bourgain's invariance of Gibbs measure

Enhanced understanding of probabilistic solution construction for 2D cubic NLS

Integration of recent random tensor estimates into classical analysis

Abstract

In a seminal paper (1996), Bourgain proved invariance of the Gibbs measure for the defocusing cubic nonlinear Schr\"odinger equation on the two-dimensional torus by constructing local-in-time solutions in a probabilistic manner. In this note, we revisit and streamline his argument, using the random tensor estimate developed by Deng, Nahmod, and Yue (2022).