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On the divergence and vorticity of vector ambit fields

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  • Sauri, Orimar

Abstract

This paper studies the asymptotic behaviour 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 Lévy basis. A full description of the rates of convergence and the limiting fields is given in terms of the roughness of the background driving Lévy 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.

Suggested Citation

  • Sauri, Orimar, 2020. "On the divergence and vorticity of vector ambit fields," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6184-6225.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:10:p:6184-6225
    DOI: 10.1016/j.spa.2020.05.007
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    References listed on IDEAS

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    1. Basse, Andreas & Pedersen, Jan, 2009. "Lévy driven moving averages and semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2970-2991, September.
    2. Jan Rataj & Luděk Zajíček, 2017. "On the structure of sets with positive reach," Mathematische Nachrichten, Wiley Blackwell, vol. 290(11-12), pages 1806-1829, August.
    3. Corcuera, José Manuel & Hedevang, Emil & Pakkanen, Mikko S. & Podolskij, Mark, 2013. "Asymptotic theory for Brownian semi-stationary processes with application to turbulence," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2552-2574.
    4. Braverman, Michael & Samorodnitsky, Gennady, 1998. "Symmetric infinitely divisible processes with sample paths in Orlicz spaces and absolute continuity of infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 78(1), pages 1-26, October.
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