Polynomial mixing for the white-forced Navier-Stokes system in the whole space

We study the mixing properties of the white-forced Navier-Stokes system in the whole space $\mathbb{R}^2$. Assuming that the noise is sufficiently non-degenerate, we prove the uniqueness of stationary measure and polynomial mixing in the dual-Lipschitz metric. The proof combines the coupling method with a Foia\c{s}-Prodi type estimate, weighted growth estimates for trajectories, and an estimate for the Leray projector involving Muckenhoupt $A_2$-class weights.