Resumen de Sub and supercritical stochastic quasi-geostrophic equation

Michael Röckner, Rongchan Zhu, Xiangchan Zhu

• 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.