Stochastic quasi-geostrophic equation (Q2890507)

The authors announce some results on 2D stochastic quasi-geostrophic equation in \(\mathbb T^{2}\) for general parameter \(\alpha \in (0, 1)\) and multiplicative noise. They prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all \(\alpha \in (0, 1)\). In the subcritical case \(\alpha > 1/2\), they prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for \(\alpha > 2/3\) provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, they prove the existence of Markov selections.