Well-Posedness and regularity properties of 2d $\beta$-plane stochastic Navier-Stokes equations in a periodic channel

We consider the 2d $\beta$-plane stochastic Navier-Stokes equations in a periodic channel. We prove the well-posedness and existence of the stationary measure, as well as certain regularity estimates concerning the support of the stationary measure. The mentioned estimates are crucial for the rigorous study of the cascade phenomena in this equation [8]. To the best of our knowledge, this is the first mathematically rigorous treatment of these equations involving both the stochastic noise and the Coriolis force.