Fractional Non-Linear, Linear and Sublinear Death Processes

Abstract

This paper is devoted to the study of a fractional version of non-linear $\mathpzc{M}^ν(t)$, $t>0$, linear $M^{\nu }(t)$, $t>0$ and sublinear ${\mathfrak{M}}^{\nu }(t)$, $t>0$ 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.

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