# Fractional Immigration-Death Processes

## TL;DR

This paper investigates explicit solutions and stochastic representations for fractional immigration-death processes, analyzing their properties and limit distributions using spectral and spectral methods.

## Contribution

It introduces a novel spectral approach to solve fractional difference-differential equations related to immigration-death processes and provides a stochastic representation for these solutions.

## Key findings

- Explicit strong solutions derived using spectral methods
- Stochastic representation via stable time-changed processes
- Limit distribution analysis of the time-changed process

## Abstract

In this paper we study explicit strong solutions for two difference-differential fractional equations, defined via the generator of an immigration-death process, by using spectral methods. Moreover, we give a stochastic representation of the solutions of such difference-differential equations by means of a stable time-changed immigration-death process and we use this stochastic representation to show boundedness and then uniqueness of these strong solutions. Finally, we study the limit distribution of the time-changed process.