Fractional Moment Theory for Anomalous Transport: A Unified Framework for Lévy Flights, Fractals, and Complex Dynamical Systems

ABSTRACT We develop a unified mathematical framework extending classical moment theory from discrete integer orders to a continuous spectrum of real orders , providing a systematic statistical characterization of complex systems exhibiting power‐law behavior. This fractional moment theory addresses the fundamental problem in anomalous transport where traditional integer moments diverge for heavy‐tailed distributions characteristic of Lévy flights, continuous time random walks, and chaotic advection. Through rigorous analysis of space‐time fractional diffusion equations with Hilfer‐composite time derivatives and Riesz‐Feller space derivatives, we establish the operator‐moment correspondence theorem proving that moments converge if and only if , where is the Lévy stability index governing asymptotic tail behavior . We derive from first principles the universal scaling law with explicit coefficient formulas expressed through Gamma functions, establishing connections to Fox H‐functions, Mittag‐Leffler relaxation, and Wright functions. Complete proofs are provided using multiple independent methods, including self‐similarity analysis, Mellin transform techniques, and asymptotic expansions. Applications are developed for turbulent dispersion obeying Richardson's four‐thirds law, Lagrangian chaos characterized by finite‐scale Lyapunov exponents, anomalous diffusion on fractal substrates, multifractal cascades, relaxation dynamics in glassy systems, epidemic spreading on scale‐free networks, and extreme value distributions. The continuous parameter enables the extraction of scaling exponents and transport coefficients from systems where variance‐based analysis fails entirely.