Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions
Department of Engineering, Aarhus University, 8000 Aarhus, Denmark
Symmetry 2020, 12(8), 1265; https://doi.org/10.3390/sym12081265
Received: 31 May 2020 / Revised: 1 July 2020 / Accepted: 13 July 2020 / Published: 1 August 2020
(This article belongs to the Special Issue Symmetry in Splitting Methods for Partial and Stochastics Differential Equations: Theory and Applications)
We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function defined as a weighted integral with respect to a Lévy basis. By construction, the stream function is homogeneous and isotropic and the corresponding vector field is, in addition, divergence-free. From a decomposition of the kernel in the Lévy-based integral, necessary conditions for the violation of parity symmetry are inferred. In particular, we focus on such fields that allow for skewness of projected increments, which is one of the cornerstones of the Kraichnan–Leith–Bachelor theory of two-dimensional turbulence.