Fractional Non-Linear, Linear and Sublinear Death Processes

TL;DR

This paper introduces fractional versions of non-linear, linear, and sublinear death processes by replacing derivatives with fractional derivatives, deriving explicit probabilities, and establishing subordination relations with fractional diffusion processes.

Contribution

It provides explicit state probabilities and probability generating functions for fractional death processes, and links them to fractional diffusion via subordination.

Findings

Explicit state probabilities derived for all three processes.

Established subordination relations with fractional diffusion processes.

Analyzed mean values and probability generating functions.

Abstract

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