Uniqueness of invariant measures for stochastic damped anisotropic Navier--Stokes equations

We study a two-dimensional Navier--Stokes system with anisotropic viscosity, linear damping term, and an additive noise on the whole space $\mathbb{R}^2$. For this model we prove uniqueness of invariant measures when the damping coefficient is sufficiently large compared to the noise intensity. The argument is based on an asymptotic coupling method and relies on anisotropic energy estimates together with exponential-type estimates for the $H^1$-energy. Since no Poincar\'e inequality is available on $\mathbb{R}^2$, the damping term is essential even for the existence of invariant measures. Our result applies to general additive noise without any non-degeneracy condition and remains valid even in the deterministic case $\sigma\equiv0$.