Fractional diffusion equation with distributed-order material derivative. Stochastic foundations

In this paper, we present the stochastic foundations of fractional dynamics driven by the fractional material derivative of distributed-order type. Before stating our main result, we present the stochastic scenario which underlies the dynamics given by the fractional material derivative. Then we introduce the Levy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with the material derivative of distributed-order type.