On Gibbs measure and weak flow for the cubic NLS with non-localised initial data

TL;DR

This paper establishes the existence of an invariant measure for the cubic nonlinear Schrödinger equation (NLS) on the real line with non-localized initial data, demonstrating solutions that preserve this measure over time.

Contribution

It proves the existence of an invariant measure supported on non-localized functions for the cubic NLS, extending the understanding of measure invariance beyond localized initial data.

Findings

Existence of an invariant measure for cubic NLS with non-localized data

Construction of solutions whose laws are invariant over time

Application of probabilistic theorems to prove measure invariance

Abstract

In this paper we prove the existence of an invariant measure for the cubic NLS $$i\partial_t u + \bigtriangleup u - |u|^2 u = 0$$ on the real line in the sense that we prove the existence of a measure $\rho$ supported by non-localised functions such that there exists random variables $X(t)$ whose laws are $\rho$ (thus independent of $t$) and such that $t\mapsto X(t)$ is a solution to the cubic NLS. Our strategy for the proof is inspired by \cite{burqtzv} and relies on the application of Prokhorov and Skorokhod Theorems to a sequence of measures which are invariant under some approximating flows, as we proved in our previous \cite{lastbaby}. However, the work by Bourgain, \cite{B00} provides a stronger result than this one, as it gives almost sure strong solutions for the cubic NLS and the invariance of the measure can be deduced from it.