Randomly Stopped Nonlinear Fractional Birth Processes

TL;DR

This paper studies nonlinear and fractional pure birth processes subordinated to various random times, deriving their distributions, differential equations, and interpretations, with applications to processes related to fractional diffusion and Mittag-Leffler functions.

Contribution

It introduces new analyses of subordinated nonlinear and fractional birth processes, including their distributions, differential equations, and interpretations, expanding understanding of processes with random time changes.

Findings

Derived state probability distributions for subordinated processes

Presented differential equations governing these processes

Analyzed compositions of fractional birth processes with random times

Abstract

We present and analyse the nonlinear classical pure birth process $\mathpzc{N} (t)$, $t>0$, and the fractional pure birth process $\mathpzc{N}^\nu (t)$, $t>0$, subordinated to various random times, namely the first-passage time $T_t$ of the standard Brownian motion $B(t)$, $t>0$, the $\alpha$-stable subordinator $\mathpzc{S}^\alpha(t)$, $\alpha \in (0,1)$, and others. For all of them we derive the state probability distribution $\hat{p}_k (t)$, $k \geq 1$ and, in some cases, we also present the corresponding governing differential equation. We also highlight interesting interpretations for both the subordinated classical birth process $\hat{\mathpzc{N}} (t)$, $t>0$, and its fractional counterpart $\hat{\mathpzc{N}}^\nu (t)$, $t>0$ in terms of classical birth processes with random rates evaluated on a stretched or squashed time scale. Various types of compositions of the fractional pure birth process $\mathpzc{N}^\nu(t)$ have been examined in the last part of the paper. In particular, the processes $\mathpzc{N}^\nu(T_t)$, $\mathpzc{N}^\nu(\mathpzc{S}^\alpha(t))$, $\mathpzc{N}^\nu(T_{2\nu}(t))$, have been analysed, where $T_{2\nu}(t)$, $t>0$, is a process related to fractional diffusion equations. Also the related process $\mathpzc{N}(\mathpzc{S}^\alpha({T_{2\nu}(t)}))$ is investigated and compared with $\mathpzc{N}(T_{2\nu}(\mathpzc{S}^\alpha(t))) = \mathpzc{N}^\nu (\mathpzc{S}^\alpha(t))$. As a byproduct of our analysis, some formulae relating Mittag--Leffler functions are obtained.