Invariant measures for the defocusing NLS

TL;DR

This paper establishes the existence and invariance of a Gibbs measure for the defocusing sub-quintic nonlinear Schrödinger equation on a 2D disc, providing insights potentially extendable to 3D.

Contribution

It introduces a Gibbs measure for the defocusing sub-quintic NLS on a 2D disc and proves its invariance, with estimates suggesting behavior in higher dimensions.

Findings

Existence of a Gibbs measure for the 2D defocusing sub-quintic NLS.

Proof of measure invariance under the NLS flow.

An estimate indicating possible dynamics in 3D.

Abstract

We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schroedinger equations on the disc of the plane $\R^2$. We also prove an estimate giving some intuition to what may happen in 3 dimensions.