We extend the Itô–Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to k-form-valued stochastic processes. The result is the Kunita–Itô–Wentzell (KIW) formula for k-forms. We also establish a correspondence between the KIW formula for k-forms derived here and a certain class of stochastic fluid dynamics models which preserve the geometric structure of deterministic ideal fluid dynamics. This geometric structure includes Eulerian and Lagrangian variational principles, Lie–Poisson Hamiltonian formulations and natural analogues of the Kelvin circulation theorem, all derived in the stochastic setting.

中文翻译：

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Kunita–Itô–Wentzell公式对随机流体动力学中k形式的影响

我们将Itô–Wentzell公式扩展为沿着半mart时的随时间变化的随机场向k形值随机过程的演化。结果是k形式的Kunita–Itô–Wentzell（KIW）公式。我们还建立了此处导出的k形式的KIW公式与某一类随机流体动力学模型之间的对应关系，该模型保留了确定性理想流体动力学的几何结构。这种几何结构包括欧拉和拉格朗日变分原理，李-泊松哈密顿公式和开尔文循环定理的自然类似物，所有这些都是在随机情况下得出的。