Operator Lévy motion and multiscaling anomalous diffusion

Abstract

The long-term limit motions of individual heavy-tailed (power-law) particle jumps that characterize anomalous diffusion may have different scaling rates in different directions. Operator stable motions $\{Y\left(t\right):t>~0\}$ are limits of d-dimensional random jumps that are scale-invariant according to $c^{H}Y\left(t\right)=Y\left(\mathrm{ct}\right),$ where H is a $d\times d$ matrix. The eigenvalues of the matrix have real parts $1/{\alpha }_j,$ with each positive ${\alpha }_j<~2.$ In each of the j principle directions, the random motion has a different Fickian or super-Fickian diffusion (dispersion) rate proportional to $t^{1/{\alpha }_j}.$ These motions have a governing equation with a spatial dispersion operator that is a mixture of fractional derivatives of different order in different directions. Subsets of the generalized fractional operator include (i) a fractional Laplacian with a single order α and a general directional mixing measure $m\left(\theta \right);$ and (ii) a fractional Laplacian with uniform mixing measure (the Riesz potential). The motivation for the generalized dispersion is the observation that tracers in natural aquifers scale at different (super-Fickian) rates in the directions parallel and perpendicular to mean flow.