Modeling and statistical analysis of non-Gaussian random fields with heavy-tailed distributions

In this paper, we investigate and develop a new approach to the numerical analysis and characterization of random fluctuations with heavy-tailed probability distribution function (PDF), such as turbulent heat flow and solar flare fluctuations. We identify the heavy-tailed random fluctuations based on the scaling properties of the tail exponent of the PDF, power-law growth of $q$th order correlation function and the self-similar properties of the contour lines in two-dimensional random fields. Moreover, this work leads to a substitution for fractional Edwards-Wilkinson (EW) equation that works in presence of $\mu$-stable L\'evy noise. Our proposed model explains the configuration dynamics of the systems with heavy-tailed correlated random fluctuations. We also present an alternative solution to the fractional EW equation in the presence of $\mu$-stable L\'evy noise in the steady-state, which is implemented numerically, using the $\mu$-stable fractional L\'evy motion. Based on the analysis of the self-similar properties of contour loops, we numerically show that the scaling properties of contour loop ensembles can qualitatively and quantitatively distinguish non-Gaussian random fields from Gaussian random fluctuations.