Moses, Noah and Joseph effects in Lévy walks

Abstract

We study a method for detecting the origins of anomalous diffusion, when it is observed in an ensemble of times-series, generated experimentally or numerically, without having knowledge about the exact underlying dynamics. The reasons for anomalous diffusive scaling of the mean-squared displacement are decomposed into three root causes: increment correlations are expressed by the 'Joseph effect' (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), fat-tails of the increment probability density lead to a 'Noah effect' (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), and non-stationarity, to the 'Moses effect' (Chen et al 2017 Phys. Rev. E 95 042141). After appropriate rescaling, based on the quantification of these effects, the increment distribution converges at increasing times to a time-invariant asymptotic shape. For different processes, this asymptotic limit can be an equilibrium state, an infinite-invariant, or an infinite-covariant density. We use numerical methods of time-series analysis to quantify the three effects in a model of a non-linearly coupled Lévy walk, compare our results to theoretical predictions, and discuss the generality of the method.