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Lévy-based spatial-temporal modelling, with applications to turbulence. (English. Russian original) Zbl 1062.60039
Russ. Math. Surv. 59, No. 1, 65-90 (2004); translation from Usp. Mat. Nauk 59, No. 1, 63-90 (2004).
This is an expository paper. It outlines several results on Ornstein-Uhlenbeck (OU) fields and stochastic intermittency (SI) fields. This is done by discussing spatial-temporal models $$X=\{X_{t}(\sigma )\}$$ and $$Y=\{Y_{t}(\sigma )\}$$ on a set $$\mathcal{S}$$ of sites $$\sigma ,$$ respectively. $$X$$ is defined by $$X_{t}(\sigma )=\int_{-\infty }^{t}\int_{ \mathcal{S}}f_{t}(\rho ,s;\sigma )Z(d\rho \times d\sigma ),$$ where $$Z$$ denotes a Lévy basis and the integrand $$f_{t}$$ is a deterministic function of special form. The SI fields are of multiplicative type, that is $$Y_{t}(\sigma )=\exp (X_{t}(\sigma )).$$ Applications of OU fields concern, among other things, dynamics of financial markets and the modelling of intensity processes for use in dynamical spatial processes of Cox type. As for SI fields, the authors are mainly interested in modelling $$n$$-point correlations and $$n$$-point correlators defined in terms of the Laplace transforms of $$X_{t}(\sigma ).$$ They sketch briefly applications such as the stochastic behaviour of turbulent energy dissipation fields viewed as continuous analogues of multiplicative cascade processes.

##### MSC:
 60G35 Signal detection and filtering (aspects of stochastic processes) 76F55 Statistical turbulence modeling 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B28 Finance etc. (MSC2000) 60E07 Infinitely divisible distributions; stable distributions 60G51 Processes with independent increments; Lévy processes
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