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Keywords:

  • Stochastic processes;
  • random walks and Levy flights;
  • econophysics;
  • isotropic turbulence.

Abstract

The meaning of non-Gaussian statistics of disordered systems is discussed. The risks of heavy tailed probability distributions of wind gusts are presented together with the discussion of an eventual power law behavior of such heavy tailed distributions. We summarize the argumentation connecting such probability distributions with Levy processes and the meaning of non-existing moments. Based on turbulence and financial market data, stochastic cascade processes are presented, which lead to anomalous statistics with heavy tails, too. Finally we show how for disordered complex systems with an underlying hierarchical structure, like the cascade structure in turbulence, a stochastic equation can be estimated directly from the data. The numerical solution of this reconstructed stochastic process gives back the anomalous statistics.