L\'evy-based spatial-temporal modelling, with applications to turbulence

O. E. Barndorff-Nielsen ; J. Schmiegel

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 1, pp. 65-90

Résumé

This paper involves certain types of spatial-temporal models constructed from Lévy bases. The dynamics is described by a field of stochastic processes $X=\{X_t(\sigma)\}$, on a set $\mathscr S$ of sites $\sigma$, defined as integrals $$ X_t(\sigma)=\int_{-\infty}^t\int_{\mathscr S}f_t(\rho,s;\sigma)\,Z(\mathrm d\rho\times\mathrm ds), $$ where $Z$ denotes a Lévy basis. The integrands $f$ are deterministic functions of the form $f_t(\rho,s;\sigma)=h_t(\rho,s;\sigma)\mathbf 1_{A_t(\sigma)}(\rho,\sigma)$, where $h_t(\rho,s;\sigma)$ has a special form and $A_t(\sigma)$ is a subset of $\mathscr S\times \mathbb R_{\leqslant t}$. The first topic is OU (Ornstein–Uhlenbeck) fields $X_t(\sigma)$, which represent certain extensions of the concept of OU processes (processes of Ornstein–Uhlenbeck type); the focus here is mainly on the potential of $X_t(\sigma)$ 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 $$ Y_t(\sigma)=\exp\{X_t(\sigma)\}. $$ This form is useful when explicitly computing expectations of the form $$ \mathsf E\{Y_{t_1}(\sigma_1)\cdots Y_{t_n}(\sigma_n)\}, $$ 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.