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The two dimensional primitive equations with multiplicative noise are studied in this paper. The existence and uniqueness of solutions in a fixed probability space and a Wentzell-Freidlin type large deviation principle for small multiplicative noise by weak convergence method are obtained.

In this paper, a small‐time large deviation principle for the stochastic non‐Newtonian fluids driven by multiplicative noise is proved. Copyright © 2016 John Wiley & Sons, Ltd.