An anomalous fractional diffusion operator

Abstract In this article, using that the fractional Laplacian can be factored into a product of the divergence operator, a Riesz potential operator and the gradient operator, we introduce an anomalous fractional diffusion operator, involving a matrix K ( x ), suitable when anomalous diffusion is being studied in a non homogeneous medium. For the case of K ( x ) a constant, symmetric positive definite matrix we show that the fractional Poisson equation is well posed, and determine the regularity of the solution in terms of the regularity of the right hand side function.