Characterizing anomalous diffusion by studying displacements

• For Lévy processes subordinated by inverse Lévy subordinators, we study the characteristic function of their increments.
• The Log-characteristic exponents of parent and subordinator of each such process determine this characteristic function.
• An operator transforms the increment characteristic function into a quantity, independent of the wave-number.
• An inverse method based on this operator finds parent and subordinator of each subordinated Lévy process.
• The method uses data sampling increment characteristic functions, quantities measured by Nuclear Magnetic Resonance.

Subordinated Lévy processes provide very diverse conceptual models for mass transport, beside other paradigms (e.g., fractional Brownian motion) generalizing Brownian motion. Some of that many models exhibit similar empirical Mean Squared Displacements growing non-linearly with time, while their increments have very different characteristic functions. In many media, such functionals can be directly measured, but accurate inversion methods adapted to them and to subordinated processes are still lacking. We show that each such process is associated to an operator that transforms the deviation from 1 of the characteristic function of its increments into a quantity that does not depend on the wave-number. We build an inversion method based on this property: it deduces the individual identity of each subordinated Lévy process from data sampling the characteristic functions of its increments.