Complete Decomposition of Anomalous Diffusion in Variable Speed Generalized Lévy Walks

Abstract

Variable Speed Generalized Lévy Walks (VGLWs) are a class of spatio-temporally coupled stochastic processes that unify a broad range of previously studied models within a single parametrized framework. Their dynamics consist of discrete random steps, or flights, during which the walker's speed varies deterministically with both the elapsed time and the total duration of the flight. We investigate the anomalous diffusive behavior of VGLWs and analyze it through decomposition into the three fundamental constitutive effects that capture violations of the Central Limit Theorem (CLT): the Joseph effect, reflecting long-range increment correlations, the Noah effect, arising from heavy-tailed step-size distributions with infinite variance, and the Moses effect, associated with statistical aging and non-stationarity. Our results show that anomalous diffusion in VGLWs is typically generated by a nontrivial combination of all three effects, rather than being attributable to a single mechanism. Strikingly, we find that within the VGLW framework the Noah exponent $L$, which quantifies the strength of the Noah effect, is unbounded from above, revealing a richer and more extreme landscape of anomalous diffusion than in previously studied Lévy-walk-type models.

22 pages, 12 figures

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