A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schr\"odinger equations

TL;DR

This paper develops a pedagogic approach to constructing invariant Gibbs measures for the 2D defocusing nonlinear Schrödinger equations on compact manifolds, demonstrating global solutions with measure-preserving properties.

Contribution

It provides a self-contained presentation of Wick renormalization using Hermite and Laguerre polynomials and constructs invariant Gibbs measures for the 2D defocusing NLS.

Findings

Construction of Gibbs measures via Wick renormalization.

Existence of global-in-time solutions with Gibbs measure initial data.

Invariance of the Gibbs measure under the flow of the NLS.

Abstract

We consider the defocusing nonlinear Schr\"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\R^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.