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 $ H^s $ with $ s\geq2-2\alpha $, where $ \alpha\in(\frac{1}{2}, 1) $. For the critical case $ \alpha = \frac12 $, we obtain the similar results in $ H^s $ with $ s>1 $.
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