Multidimensional fractional advection-dispersion equations and related stochastic processes

In this paper we study multidimensional fractional advection-dispersion equations involving fractional directional derivatives both from a deterministic and a stochastic point of view. For such equations we show the connection with a class of multidimensional Levy processes. We introduce a novel Levy-Khinchine formula involving fractional gradients and study the corresponding infinitesimal generator of multi-dimensional random processes. We also consider more general fractional transport equations involving Frobenius-Perron operators and their stochastic solutions. Finally, some results about fractional power of second order directional derivatives and their applications are also provided.