Fractional Non-Linear, Linear and Sublinear Death Processes

This paper is devoted to the study of a fractional version of non-linear M-v(t) t > 0, linear M-v(t), t > 0 and sublinear M-v(t), t > 0 and death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan-Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process T (2 nu)(t), t > 0. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.