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Lévy-based spatial-temporal modelling, with applications to turbulence
O. E. Barndorff-Nielsen,
J. Schmiegel University of Aarhus
Abstract:
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.
UDC:
519.248:53
MSC: Primary
60G35,
76F55; Secondary
62P05,
91B28,
60E07,
60G51,
60G57,
60G60 Received: 20.06.2003
DOI:
10.4213/rm701
Fulltext:
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