Invariant Gibbs measures for 1D NLS in a trap

TL;DR

This paper constructs and proves the invariance of Gibbs measures for the 1D cubic nonlinear Schrödinger equation with a trapping potential, establishing global well-posedness and measure invariance almost surely.

Contribution

It introduces new Gibbs measures for the NLS with trapping potential and proves their invariance, extending the understanding of measure-preserving dynamics in this setting.

Findings

Gibbs measures are constructed for the 1D NLS with trapping potential.

The Cauchy problem is shown to be globally well-posed almost surely on the measure support.

Gibbs measures are proven to be invariant under the NLS flow.

Abstract

We consider the one dimensional cubic nonlinear Schr{\"o}dinger equation with trapping potential behaving like |x| s (s > 1) at infinity. We construct Gibbs measures associated to the equation and prove that the Cauchy problem is globally well-posed almost surely on their support. Consequently, the Gibbs measure is indeed invariant under the flow of the equation. We also address the construction and invariance of canonical Gibbs measures, conditioned on the L 2 mass.