Decomposition of anomalous diffusion in generalized Lévy walks into its constitutive effects

Anomalous diffusion is observed in a variety of physical and social systems, including blinking quantum dots, animal locomotion, intra-day trades in financial markets and cold atoms in dissipative optical lattices. Generalized Lévy walks can be used to model their dynamics. We show that the anomalous diffusive behavior found in these systems can be decomposed into three fundamental constitutive causes. These causes, or effects, are related to ways that the Central Limit Theorem fails. The increments generated through the stochastic process can have either long-time correlations, infinite variance, or be non-stationary. Each of these properties can cause anomalous diffusion and is characterized by what is known as the Joseph, Noah and Moses effects, respectively. In generalized Lévy walks, a complex combination of these effects leads to the observed sub- and super-diffusive behaviors. We analytically calculate the scaling exponents determining each of the three constitutive effects and confirm the results with numerical simulations. The results satisfy a fundamental scaling relation between the exponents.