Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay

In this paper, we investigate a stochastic fractionally dissipative quasi-geostrophic equation driven by a multiplicative white noise, whose external forces contain hereditary characteristics. The existence and uniqueness of both local martingale and local pathwise solutions are established in \begin{document}$ H^s $\end{document} with \begin{document}$ s\geq2-2\alpha $\end{document} , where \begin{document}$ \alpha\in(\frac{1}{2}, 1) $\end{document} . For the critical case \begin{document}$ \alpha = \frac12 $\end{document} , we obtain the similar results in \begin{document}$ H^s $\end{document} with \begin{document}$ s>1 $\end{document} .