All order moments and other functionals of the increments of some non-Markovian processes

We propose a theoretical framework to analyze nuclear magnetic resonance (NMR) experiments for the description of dispersion processes featuring memory effects. Memory effects, addressed here, can be represented by subordinated Brownian motions with random time changes that invert Lévy time processes, with stable densities of exponent between 0 and 1. According to whether the Lévy process has a drift equal to zero or not, the subordinated motion has a p.d.f that solves the fractional Fokker–Planck equation or the fractal mobile/immobile model. NMR experiments can measure the characteristic function of displacements of water molecules and facilitate their interpretation in media showing memory effects. We give mathematical expressions for the moments and averaged exponentials of the increment of subordinated Brownian motions within the framework of fractal MIM and FFPE. The results are illustrated on the basis of a numerical method