This paper studies the asymptotic behavior of the flux and circulation of a subclass of random fields within the family of 2-dimensional vector ambit fields. We show that, under proper normalization, the flux and the circulation converge stably in distribution to certain stationary random fields that are defined as line integrals of a L\'evy basis. A full description of the rates of convergence and the limiting fields is given in terms of the roughness of the background driving L\'evy basis and the geometry of the ambit set involved. We further discuss the connection of our results with the classical Divergence and Vorticity Theorems. Finally, we introduce a class of models that are capable to reflect stationarity, isotropy and null divergence as key properties.

中文翻译：

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关于矢量域场的散度和涡度

本文研究了二维矢量范围场族中随机场子类的通量和环流的渐近行为。我们表明，在适当的归一化下，通量和环流在分布中稳定收敛到某些定义为 L'evy 基的线积分的平稳随机场。根据背景驱动 L'evy 基的粗糙度和所涉及的范围集的几何形状，给出了收敛速度和限制场的完整描述。我们进一步讨论了我们的结果与经典散度和涡度定理的联系。最后，我们介绍了一类能够反映平稳性、各向同性和零散度作为关键属性的模型。