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 Lévy processes. We introduce a novel Lévy-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.
"Multidimensional fractional advection-dispersion equations and related stochastic processes." Electron. J. Probab. 19 1 - 31, 2014. https://doi.org/10.1214/EJP.v19-2854