Invariant measures of Gaussian type for 2D turbulence

Gaussian measures μβ,ν are associated to some stochastic 2D models of turbulence. They are Gibbs measures constructed by means of an invariant quantity of the system depending on some parameter β (related to the 2D nature of the fluid) and the viscosity ν. We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure μβ,ν is invariant for this flow and is the unique invariant measure. Finally, we prove that the deterministic inviscid equation has a μβ,ν-stationary solution (for any ν >0).