Catalytic branching processes via spine techniques and renewal theory

TL;DR

This paper advances the understanding of catalytic branching processes by applying spine techniques and renewal theory to analyze moments and long-term behavior, resulting in a system of PDEs for particle counts.

Contribution

It introduces a novel application of spine techniques and renewal theory to derive PDE systems for catalytic branching processes, enhancing moment analysis methods.

Findings

Derived PDE system for particle counts in catalytic media

Established long-time behavior using renewal theorems

Enhanced moment analysis of branching processes

Abstract

In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behaviour is then deduced from renewal theorems and induction.