We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane ${ℝ}^2$. We also prove an estimate giving some intuition to what may happen in $3$ dimensions.

The Ito equation is shown to be a geodesic flow of $L^2$ metric on the semidirect product space $\widehat{{\mathrm{𝐷𝑖𝑓𝑓}}^s\left(S^1\right)⨀C^{\infty }\left(S^1\right)}$, where ${\mathrm{𝐷𝑖𝑓𝑓}}^s\left(S^1\right)$ is the group of orientation preserving Sobolev $H^s$ diffeomorphisms of the circle. We also study a geodesic flow of a $H^1$ metric.