We consider the 2d \(\beta \) -plane stochastic Navier–Stokes equations in a periodic channel. We prove the well-posedness and the existence of the stationary measure, as well as certain regularity estimates concerning the support of the stationary measure. The mentioned explicit estimates are crucial for the rigorous study conducted in [8] of cascade phenomena for these equations. Although one may be able to invoke a more abstract theory to claim well-posedness for the associated initial value, to the best of our knowledge, this is the first mathematically rigorous and explicit treatment of this problem involving both the stochastic noise and the Coriolis force in a periodic channel.
