2D hydrodynamical systems: invariant measures of Gaussian type

Abstract

Gaussian measures $μ^{β,ν}$ 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 $β$ (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 $μ^{β,ν}$ is invariant for this flow and is unique. Finally, we prove that the deterministic inviscid equation has a $μ^{β,ν}$-stationary solution (for any $ν>0$).