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In this paper, we prove that the solution constructed in \cite{BR16}satisfies the stochastic vorticity equations with the stochastic integrationbeing understood in the sense of the integration of controlled rough pathintroduced in \cite{G04}. As a result, we obtain the existence and uniquenessof the global solutions to the stochastic vorticity equations in 3D case forthe small initial data independent of time, which can be viewed as a stochasticversion of the Kato-Fujita result (see \cite{KF62}).

A remark on global solutions to random 3D vorticity equations for small initial data