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Our first purpose is to extend the results from \cite{T} on the radialdefocusing NLS on the disc in $\mathbb{R}^2$ to arbitrary smooth (defocusing)nonlinearities and show the existence of a well-defined flow on the support ofthe Gibbs measure (which is the natural extension of the classical flow forsmooth data). We follow a similar approach as in \cite{BB-1} exploiting certainadditional a priori space-time bounds that are provided by the invariance ofthe Gibbs measure.  Next, we consider the radial focusing equation with cubic nonlinearity (themass-subcritical case was studied in \cite{T2}) where the Gibbs measure issubject to an $L^2$-norm restriction. A phase transition is established, of thesame nature as studied in the work of Lebowitz-Rose-Speer \cite{LRS} on thetorus. For sufficiently small $L^2$-norm, the Gibbs measure is absolutelycontinuous with respect to the free measure, and moreover we have awell-defined dynamics.

Almost sure global well posedness for the radial nonlinear Schrodinger equation on the unit ball I: the 2D case