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We extend the It\^o-Wentzell formula for the evolution of a time-dependentstochastic field along a semimartingale to $k$-form-valued stochasticprocesses. The result is the Kunita-It\^o-Wentzell (KIW) formula for $k$-forms.We also establish a correspondence between the KIW formula for $k$-formsderived here and a certain class of stochastic fluid dynamics models whichpreserve the geometric structure of deterministic ideal fluid dynamics. Thisgeometric structure includes Eulerian and Lagrangian variational principles,Lie--Poisson Hamiltonian formulations and natural analogues of the Kelvincirculation theorem, all derived in the stochastic setting.

Implications of Kunita-It\^o-Wentzell formula for $k$-forms in stochastic fluid dynamics