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

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.