  In this note we study the 2d stochastic quasi-geostrophic equation in
$\mathbb{T}^2$ for general parameter $\alpha\in (0,1)$ and multiplicative
noise. We 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 $\alpha>1/2$, we 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, we prove the
existence of Markov selections.

Probability