Global well posedness and ergodic results in regular Sobolev spaces for the nonlinear Schrödinger equation with multiplicative noise and arbitrary power of the nonlinearity

We consider the nonlinear Schrödinger equation on the $d$-dimensional torus ${\mathbb{T}}^d$, with the nonlinearity of polynomial type $\mathrm{\mid }u{\mathrm{\mid }}^{2\sigma }u$. For any $\sigma \in \mathbb{N}$ and $s>\frac{d}{2}$ we prove that adding to this equation a suitable stochastic forcing term there exists a unique global solution for any initial data in $H^s({\mathbb{T}}^d)$. The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting. Moreover we prove existence of invariant measures and their uniqueness under more restrictive assumptions on the noise term.