Probabilistic well-posedness and Gibbs measure evolution for the non linear Schrödinger equation on the two dimensional sphere

We present recent results on probabilistic well-posedness of the two dimensional NLS, posed on the sphere. These results deal with low regularity solutions. The construction of such solutions is beyond the scope of applicability of the deterministic methods of Burq-Gérard-Tzvetkov developed between 2000 and 2004.