On the Divergence and Vorticity of Vector Ambit Fields

TL;DR

This paper investigates the asymptotic behavior of flux and circulation in 2D vector ambit fields, revealing their convergence to stationary random fields defined by Lévy basis integrals, with implications for divergence and vorticity theorems.

Contribution

It provides a detailed analysis of the convergence rates and limiting behavior of flux and circulation in vector ambit fields, linking them to Lévy basis properties and geometric factors.

Findings

Flux and circulation converge to stationary random fields under normalization

Limiting fields are characterized as line integrals of Lévy bases

Results connect to classical divergence and vorticity theorems

Abstract

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.