Fractional derivatives of multidimensional Colombeau generalized stochastic processes

Abstract

We consider fractional derivatives of a Colombeau generalized stochastic process G defined on ℝ n . We first introduce the Caputo fractional derivative of a one-dimensional Colombeau generalized stochastic process and then generalize the procedure to the Caputo partial fractional derivatives of a multidimensional Colombeau generalized stochastic process. To do so, the Colombeau generalized stochastic process G has to have a compact support. We prove that an arbitrary Caputo partial fractional derivative of a compactly supported Colombeau generalized stochastic process is a Colombeau generalized stochastic process itself, but not necessarily with a compact support. © 2013 Versita Warsaw and Springer-Verlag Wien.