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

Jules Olayé, Milica Tomasevic

TL;DR

This work analyzes the long-time behavior of a multidimensional age-structured branching process with a singular, non-absolute continuous jump kernel modeling telomere shortening. By combining stochastic comparisons with Bellman-Harris renewal theory and a weak form of Harnack inequality, the authors establish exponential ergodicity of the first moment semigroup after an exponential normalisation, despite an unbounded birth rate and a non-compact kernel. A weighted-normalised semigroup and an auxiliary Markov process are constructed to obtain a precise stationary profile $(N,oldsymbol{ u},oldsymbol{ ho})$, along with a density representation that separates telomere-length and age components. The paper provides concrete, verifiable conditions under which the ergodic result holds and illustrates two models where the assumptions are satisfied, linking the mathematics closely to telomere biology and suggesting applicability to other age-structured branching systems. Overall, the results offer rigorous verification of stationary telomere-length distributions in complex, biologically realistic branching processes and supply a framework for exploring further Lyapunov-type conditions and large-number limits in such systems.

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.

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

TL;DR

This work analyzes the long-time behavior of a multidimensional age-structured branching process with a singular, non-absolute continuous jump kernel modeling telomere shortening. By combining stochastic comparisons with Bellman-Harris renewal theory and a weak form of Harnack inequality, the authors establish exponential ergodicity of the first moment semigroup after an exponential normalisation, despite an unbounded birth rate and a non-compact kernel. A weighted-normalised semigroup and an auxiliary Markov process are constructed to obtain a precise stationary profile , along with a density representation that separates telomere-length and age components. The paper provides concrete, verifiable conditions under which the ergodic result holds and illustrates two models where the assumptions are satisfied, linking the mathematics closely to telomere biology and suggesting applicability to other age-structured branching systems. Overall, the results offer rigorous verification of stationary telomere-length distributions in complex, biologically realistic branching processes and supply a framework for exploring further Lyapunov-type conditions and large-number limits in such systems.

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.
Paper Structure (106 sections, 34 theorems, 304 equations)

This paper contains 106 sections, 34 theorems, 304 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Telomere shortening.
  4. Stationary profile.
  5. Main result, literature and proof strategy
  6. Main result.
  7. Difficulties.
  8. Literature.
  9. Proof strategy.
  10. Example of a model
  11. Organisation of the paper
  12. Notations, preliminaries and main result
  13. Notations and first definitions
  14. $(N_1)\!:$ Probability space and related notations.
  15. $(N_2)\!:$ Trait space and Ulam-Harris-Neveu notation.
  16. ...and 91 more sections

Key Result

Theorem 2.7

Let us assume that eq:birth_rate_assumption and eq:uniform_random_variables_assumption hold, and that the variable $Y_0$ defined in Notation notation:initial_condition_process verifies $\mathbb{E}\left[Y_0(1)\right] < +\infty$. Then, there exists a $\left(\mathcal{F}_t\right)_{t\geq0}$-adapted càdlà

Theorems & Definitions (60)

  • Example 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7: Well-posedness and non-explosion
  • Definition 2.8: Measure solution to \ref{['eq:PDE_model_intro']}
  • Theorem 2.9: Deterministic representation of \ref{['eq:SDE_model_intro']}
  • Theorem 2.10: Main result
  • ...and 50 more