Invariant Gibbs measures for the one-dimensional quintic nonlinear Schr\"odinger equation in infinite volume

TL;DR

This paper proves the invariance of Gibbs measures for the one-dimensional defocusing quintic nonlinear Schrödinger equation on the real line, extending previous cubic case results using stochastic quantization techniques.

Contribution

It introduces a new growth estimate for infinite-volume measures and establishes invariance for the quintic NLS, advancing understanding of infinite-volume Gibbs measures.

Findings

Invariance of Gibbs measure for the quintic NLS on the real line.

Development of a growth estimate for infinite-volume measures.

Application of stochastic quantization to infinite-volume setting.

Abstract

We prove the invariance of the Gibbs measure for the defocusing quintic nonlinear Schr\"odinger equation on the real line. This builds on earlier work by Bourgain, who treated the cubic nonlinearity. The key new ingredient is a growth estimate for the infinite-volume $\Phi^{p+1}_1$-measures, which is proven via the stochastic quantization method.