Operator Levy motion and multiscaling anomalous diffusion

The lone-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(t):t greater than or equal to0} are limits of d-dimensional random jumps that are scale-invariant according to (CY)-Y-H(t) = Y(ct), where H is a d x d matrix. The eigenvalues of the matrix have real parts 1/alpha (j), with each positive alpha (j)less than or equal to2. In each of the j principle directions, the random motion has a different Fickian or super-Fickian diffusion (dispersion) rate proportional to t(1/alphaj). 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 alpha and a general directional mixing measure m(theta); 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.