Invariant Gibbs measures and global strong solutions for nonlinear Schr\"odinger equations in dimension two

TL;DR

This paper proves almost sure global well-posedness and invariance of the Gibbs measure for the defocusing nonlinear Schrödinger equation on a two-dimensional torus, introducing random averaging operators as a key tool.

Contribution

It establishes the invariance of Gibbs measures and global solutions for 2D nonlinear Schrödinger equations using a novel approach with random averaging operators.

Findings

Almost sure global well-posedness with respect to Gibbs measure

Invariance of Gibbs measure under the flow

Introduction of random averaging operators as a new technique

Abstract

We consider the defocusing nonlinear Schr\"odinger equation on $\mathbb{T}^2$ with Wick ordered power nonlinearity, and prove almost sure global well-posedness with respect to the associated Gibbs measure. The heart of the matter is the uniqueness of the solution as limit of solutions to canonically truncated systems. The invariance of the Gibbs measure under the global dynamics follows as a consequence. The proof relies on the novel idea of random averaging operators.