Levy-based spatial-temporal modelling, with applications to turbulence

This paper involves certain types of spatial-temporal models constructed from Levy bases. The dynamics is described by a field of stochastic processes X={Xt(σ)}, on a set S of sites σ, defined as integrals Xt(σ)=∫-∞t∫Sft(ρ,s;σ) Z(dρxds), where Z denotes a Levy 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) has a special form and At(σ) is a subset of SxR≤t. The first topic is OU (Ornstein-Uhlenbeck) fields Xt(σ), which represent certain extensions of the concept of OU processes (processes of Ornstein-Uhlenbeck type); the focus here is mainly 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 modelling of spatial-temporal correlations of SI (stochastic intermittency) fields of the form Yt(σ=exp{Xt(σ)}. This form is useful when explicitly computing expectations of the form E{Yt1(σ1)cYtn(σn)}, which are used to characterize correlations. The SI fields can be viewed as a dynamical, continuous, and homogeneous generalization of turbulent cascades. In this connection an SI field is constructed with spatial-temporal scaling behaviour that agrees with the energy dissipation observed in turbulent flows. Some parallels of this construction are also briefly sketched