2D hydrodynamical systems: invariant measures of Gaussian type

Gaussian measures ${\mu }^{\beta ,\nu }$ are associated to some stochastic 2D hydrodynamical systems. They are of Gibbsian type and are constructed by means of some invariant quantities of the system depending on some parameter $\beta$ (related to the 2D nature of the fluid) and the viscosity $\nu$. We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure ${\mu }^{\beta ,\nu }$ is invariant for this flow and is unique. Finally, we prove that the deterministic inviscid equation has a ${\mu }^{\beta ,\nu }$-stationary solution (for any $\nu >0$).