Lévy-based Tempo-Spatial Modelling; with Applications to Turbulence

This paper discusses certain types of tempo-spatial models constructed from Lévy bases. The dynamics are described by a field of stochastic processes X = {Xt(σ)}, on a set S of sites σ, defined as integrals Xt(σ) = ∫ t −∞ ∫ S ft(ρ, s;σ)Z(dρ× ds) where Z denotes a Lévy basis. The integrands f are deterministic functions of the form ft(ρ, s;σ) = ht(ρ, s;σ)1At(σ)(ρ, σ) where ht(ρ, s;σ) is of a special kind and At(σ) is a subset of S × R≤t. We first consider OU (Ornstein-Uhlenbeck) fields Xt(σ) representing several extensions of the concept of OU processes (processes of Ornstein-Uhlenbeck type), with the main focus on the potential of Xt(σ) for dynamic modelling. Applications to dynamical spatial processes of Cox type are briefly indicated. The second part of the paper discusses the modelling of tempo-spatial correlations of SI (stochastic intermittency) fields of the form Yt(σ) = exp {Xt(σ)} . This form allows for explicit calculation of expectations E{Yt1(σ1) · · · Ytn(σn)}, which we use to characterise correlations. SI fields can be thought of as a dynamical, continuous and homogeneous generalisation of turbulent cascades. In this respect we construct an SI field with tempo-spatial scaling behaviour that accords with the energy dissipation observed in turbulent flows. Some parallels of this construction are also briefly sketched.