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This article is cited in 71 scientific papers (total in 71 papers)
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.
Received: 20.06.2003
Citation:
O. E. Barndorff-Nielsen, J. Schmiegel, “Lévy-based spatial-temporal modelling, with applications to turbulence”, Uspekhi Mat. Nauk, 59:1(355) (2004), 63–90; Russian Math. Surveys, 59:1 (2004), 65–90
Linking options:
https://www.mathnet.ru/eng/rm701https://doi.org/10.1070/RM2004v059n01ABEH000701 https://www.mathnet.ru/eng/rm/v59/i1/p63
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Abstract page: | 1531 | Russian version PDF: | 290 | English version PDF: | 18 | References: | 91 | First page: | 1 |
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