Modelling Framework
Barndorff-Nielsen, O.E. and Schmiegel, J. (2009): Brownian semistationary processes and volatility/intermittency. In Albrecher, H., Rungaldier, W. and Schachermeyer, W. (Eds.): Advanced Financial Modelling. Radon Series Comp. Appl. Math. 8. Pp. 1-26.
Barndorff-Nielsen, O.E. and Schmiegel, J. (2008): A stochastic differential equation framework for the turbulent velocity field. Theory of Probability and its Applications 52, 372-388.
Barndorff-Nielsen, O.E. and Schmiegel, J. (2007): Ambit processes; with applications to turbulence and cancer growth. In Benth, F.E., Nunno, G.D., Linstrøm, T., Øksendal, B., and Zhang, T. (Eds.): Stochastic Analysis and Applications: The Abel Symposium 2005. Heidelberg: Springer. Pp. 93-124.
Barndorff-Nielsen, O.E. and Schmiegel, J. (2003): Lévy-based tempo-spatial modelling; with applicacations to turbulence. Uspekhi Mat. Nauk 159, 65-91.
Phenomenology
Barndorff-Nielsen, O.E. and Schmiegel, J. (2008): Time change, volatility and turbulence. In Sarychev, A., Shiryaev, A., Guerra, M. and Grossinho, M.d.R. (Eds.): Proceedings of the Workshop on Mathematical Control Theory and Finance. Lisbon 2007. Berlin: Springer. Pp. 29-53.
Barndorff-Nielsen, O.E., Blæsild, P., and Schmiegel, J. (2004): A parsimonious and universal description of turbulent velocity increments. Eur. Phys. J. B 41, 345-363.
Intermittency/Volatility
Cleve, J., Schmiegel, J., and Greiner, M. (2008): Apparent scale correlations in a random multifractal process. Europ. Phys. J. B 63, 109-116.
Schmiegel, J., Barndorff-Nielsen, O.E. and Eggers, H.C. (2006): A class of spatio-temporal and causal stochastic processes, with application to multiscaling and multifractality. South African Journal of Science 101, 513-519.
Schmiegel, J. (2005): Self-scaling of turbulent energy dissipation correlators. Phys. Lett. A 337, 342-353.
Schmiegel, J., Cleve, J., Eggers, H.C., Pearson, B.R., and Greiner, M. (2004): Stochastic energy-cascade model for (1+1) dimensional fully developed turbulence. Phys. Lett. A 320, 247-253.
models the velocity vector as a function of space and time. It seems to be enough to choose the subordination
is white noise in space-time. With this choice of model ingredients the second term in the above formula controls the skewness of velocity increments, and the decompostion of the velocity fields into two terms resembles the well known Reynolds decomposition into a slowly varying term (the skewness term) and a rapidly varying term (the Brownian stochastic integral).
