Sub and supercritical stochastic quasi-geostrophic equation

In this paper, we study the 2D stochastic quasi-geostrophic equation on T2 for general parameter ..(0,1) and multiplicative noise.We prove the existence of weak solutions and Markov selections for multiplicative noise for all ..(0,1). In the subcritical case .>1/2, we prove existence and uniqueness of (probabilistically) strong solutions.Moreover, we prove ergodicity for the solution of the stochastic quasi-geostrophic equations in the subcritical case driven by possibly degenerate noise.The law of large numbers for the solution of the stochastic quasi-geostrophic equations in the subcritical case is also established.In the case of nondegenerate noise and .>2/3 in addition exponential ergodicity is proved