Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel modelling telomere shortening

TL;DR

This paper analyzes the long-term behavior of a complex multidimensional age-dependent branching process with a singular jump kernel, inspired by telomere shortening in cell populations, demonstrating exponential ergodicity despite unbounded rates.

Contribution

It introduces a novel multidimensional model with age-dependent branching and singular kernels, proving exponential ergodicity under challenging conditions.

Findings

Established exponential ergodicity with exponential normalization.

Handled unbounded birth rates via comparison to Bellman-Harris processes.

Applied a weak Harnack inequality to manage non-compact transition kernels.

Abstract

In this article, we investigate the ergodic behaviour of a multidimensional age-dependent branching process with a singular jump kernel, motivated by studying the phenomenon of telomere shortening in cell populations. Our model tracks individuals evolving within a continuous-time framework indexed by a binary tree, characterised by age and a multidimensional trait. Branching events occur with rates dependent on age, where offspring inherit traits from their parent with random increase or decrease in some coordinates, while the most of them are left unchanged. Exponential ergodicity is obtained at the cost of an exponential normalisation, despite the fact that we have an unbounded age-dependent birth rate that may depend on the multidimensional trait, and a non-compact transition kernel. These two difficulties are respectively treated by stochastically comparing our model to Bellman-Harris processes, and by using a weak form of a Harnack inequality. We conclude this study by giving examples where the assumptions of our main result are verified.