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We consider the incompressible 2D Navier-Stokes equations with periodicboundary conditions driven by a deterministic time periodic forcing and adegenerate stochastic forcing. We show that the system possesses a uniqueergodic periodic invariant measure which is exponentially mixing under aWasserstein metric. We also prove the weak law of large numbers for thecontinuous time inhomogeneous solution process. In addition, we obtain the weaklaw of large numbers and central limit theorem by restricting the inhomogeneoussolution process to periodic times. The results are independent of the strengthof the noise and hold true for any value of viscosity with a lower bound$\nu_1$ characterized by the Grashof number $G_1$ associated with thedeterministic forcing. In the laminar case, there is a larger lower bound$\nu_2$ of the viscosity characterized by the Grashof number $G_2$ associatedwith both the deterministic and random forcing. We prove that in this laminarcase, the system has trivial dynamics for any viscosity larger than $\nu_2$ bydemonstrating the existence of a unique globally exponentially stable randomperiodic solution that supports the unique periodic invariant measure.

Statistical Properties of 2D Stochastic Navier-Stokes Equations with Time-Periodic Forcing and Degenerate Stochastic Forcing