Sub and supercritical stochastic Quasi-geostrophic equation1

In this paper, we study the 2D stochastic quasi-geostrophic equation on T² for general parameter α ∈ (0, 1) and multiplicative noise.We prove the existence of weak solutions and Markov selections for multiplicative noise for all α ∈ (0, 1). In the subcritical case α > 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 α > 2/3 in addition exponential ergodicity is proved.