The primitive equations with stochastic wind driven boundary conditions

The primitive equations for geophysical flows are studied under the influence of stochastic wind driven boundary conditions modeled by a cylindrical Wiener process. We adapt an approach by Da Prato and Zabczyk for stochastic boundary value problems to define a notion of solutions. Then a rigorous treatment of these stochastic boundary conditions, which combines stochastic and deterministic methods, yields that these equations admit a unique, local pathwise solution within the anisotropic $L_t^q$-$H_z^{-1,p}{L}_{xy}^p$-setting. This solution is constructed in critical spaces.