Multi-type branching processes with immigration generated by point processes

TL;DR

This paper explores advanced branching processes with immigration driven by complex point processes, extending previous models from Poisson to determinantal point processes to better understand their behavior.

Contribution

It introduces a framework for analyzing branching processes with immigration generated by general point processes, moving beyond traditional Poisson models.

Findings

Demonstrates how to adapt techniques from Poisson to determinantal point process immigration.

Provides a methodology for analyzing branching processes with complex immigration mechanisms.

Extends theoretical understanding of processes with non-homogeneous and determinantal immigration.

Abstract

Following the pivotal work of Sevastyanov, who considered branching processes with homogeneous Poisson immigration, much has been done to understand the behaviour of such processes under different types of branching and immigration mechanisms. Recently, the case where the times of immigration are generated by a non-homogeneous Poisson process was considered in depth. In this work, we try to demonstrate how one can use the framework of point processes in order to go beyond the Poisson process. As an illustration, we show how to transfer techniques from the case of Poisson immigration to the case where it is spanned by a determinantal point process.