Invariant Gibbs measures for the one-dimensional quintic nonlinear Schrödinger equation in infinite volume

We prove the invariance of the Gibbs measure for the defocusing quintic nonlinear Schrödinger 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 ${\mathrm{\Phi }}_1^{p+1}$-measures, which is proven via the stochastic quantization method.