We show the existence and  uniqueness of a local maximal solution to stochastic nonlinear Schr\"odinger equations with multiplicative noise on a compact d-dimensional  Riemannian manifold. Our  approach uses a stochastic version of the Strichartz inequalities.

When d=2, we then give sufficient conditions to have a global solution in the focusing and defocusing cases. This is based on conservation laws; to prove them with a random perturbation, we need to  reformulate our NLS equation in terms of a Stratonovich integral, which requires more regularity on the driving noise and on the diffusion coefficient.

This is joint work with Z. Brzezniak.