Probabilistic well-posedeness for the nonlinear Schrödinger equation on the $2d$ sphere I: positive regularities

We establish the probabilistic well-posedness of the nonlinear Schrödinger equation on the $2d$ sphere ${\mathbb{S}}^2$. The initial data are distributed according to Gaussian measures with typical regularity $H^s({\mathbb{S}}^2)$, for $s>0$. This level of regularity goes significantly beyond existing deterministic results, in a regime where the flow map cannot be extended uniformly continuously.