We discuss two methods for establishing the invariance of the white noise for the periodic KdV. First, we briefly go over the basic theory of Gaussian Hilbert spaces and abstract Wiener spaces and show that the Fourier-Lebesgue space $\mathcal{F}L^{s, p}$ captures the regualrity of the white noise for $sp < -1$. We then establish local well-posedness (LWP) of KdV via the second iteration introduced by Bourgain. This in turn provides almost sure global well-posedness (GWP) of KdV as well as the invariance of the white noise. Then, we discuss how one can use the same idea to obtain LWP of the stochastic KdV with the additive space-time white noise in the periodic setting.Next, we consider the weak convergence problem of the grand canonical ensemble (i.e. the interpolation measure of the usual Gibbs measure and the white noise) with a small parameter (tending to 0) to the white noise. This result, combined with the GWP in $H^{-1}$ by Kappeler and Topalov, provides another proof of the invariance of the white noise for KdV. In this talk, we discuss the same weak convergence problem for mKdV and cubic NLS, which provides the "formal'' invariance of the white noise. This part is a joint work with J. Quastel and B. Valko.
Lastly, if time permits, we discuss well-posedness of the Wick ordered cubic NLS on the Gaussian ensembles below $L2$. The main ingredient is nonlinear smoothing under randomization of initial data. For GWP, we also use the invariance (of the Gaussian ensemble) under the linear flow. This part is a joint work with J. Colliander.