Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A quasi-stationary approach to the long-term asymptotics of the growth-fragmentation equation

Denis Villemonais, Alexander Watson

TL;DR

This work develops a unified quasi-stationary framework for the long-term behavior of the growth-fragmentation equation. By linking the evolution to a sub-Markov process through an h-transform and a Lyapunov function, it proves existence and uniqueness of the associated semigroup and establishes exponential convergence to an asymptotic profile with a spectral gap. The approach accommodates broad growth-fragmentation models, including self-similar kernels and mass-conserving or entrance/exit boundary behaviors, and yields tractable criteria for geometric convergence. The results advance the theory by providing a robust probabilistic representation, explicit asymptotic forms, and practical conditions to verify in common biological and physical settings, thereby offering a unified and powerful toolkit for analyzing long-time dynamics in fragmentation systems.

Abstract

In a growth-fragmentation system, cells grow in size slowly and split apart at random. Typically, the number of cells in the system grows exponentially and the distribution of the sizes of cells settles into an equilibrium 'asymptotic profile'. In this work we introduce a new method to prove this asymptotic behaviour for the growth-fragmentation equation, and show that the convergence to the asymptotic profile occurs at exponential rate. We do this by identifying an associated sub-Markov process and studying its quasi-stationary behaviour via a Lyapunov function condition. By doing so, we are able to simplify and generalise results in a number of common cases and offer a unified framework for their study. In the course of this work we are also able to prove the existence and uniqueness of solutions to the growth-fragmentation equation in a wide range of situations.

A quasi-stationary approach to the long-term asymptotics of the growth-fragmentation equation

TL;DR

This work develops a unified quasi-stationary framework for the long-term behavior of the growth-fragmentation equation. By linking the evolution to a sub-Markov process through an h-transform and a Lyapunov function, it proves existence and uniqueness of the associated semigroup and establishes exponential convergence to an asymptotic profile with a spectral gap. The approach accommodates broad growth-fragmentation models, including self-similar kernels and mass-conserving or entrance/exit boundary behaviors, and yields tractable criteria for geometric convergence. The results advance the theory by providing a robust probabilistic representation, explicit asymptotic forms, and practical conditions to verify in common biological and physical settings, thereby offering a unified and powerful toolkit for analyzing long-time dynamics in fragmentation systems.

Abstract

In a growth-fragmentation system, cells grow in size slowly and split apart at random. Typically, the number of cells in the system grows exponentially and the distribution of the sizes of cells settles into an equilibrium 'asymptotic profile'. In this work we introduce a new method to prove this asymptotic behaviour for the growth-fragmentation equation, and show that the convergence to the asymptotic profile occurs at exponential rate. We do this by identifying an associated sub-Markov process and studying its quasi-stationary behaviour via a Lyapunov function condition. By doing so, we are able to simplify and generalise results in a number of common cases and offer a unified framework for their study. In the course of this work we are also able to prove the existence and uniqueness of solutions to the growth-fragmentation equation in a wide range of situations.
Paper Structure (26 sections, 32 theorems, 374 equations)

This paper contains 26 sections, 32 theorems, 374 equations.

Table of Contents

  1. Introduction
  2. Existence of a unique solution to the growth-fragmentation equation
  3. An auxiliary Markov process
  4. Uniqueness of a Markov semigroup generated by L
  5. Technical lemmas
  6. Representation of the semigroup Q by a càdlàg Markov process
  7. Conclusion of the proof of Theorem \ref{['thm:semigroup']}
  8. Proof of Corollary \ref{['cor:1']}
  9. Proof of Corollary \ref{['cor:consistence']}
  10. Long time asymptotics of the solution to the growth-fragmentation equation
  11. Applications
  12. Entrance boundary
  13. Pseudo-entrance boundary and mass conservation
  14. Critical case, s comparable to ln x
  15. Critical case, K comparable to a constant
  16. ...and 11 more sections

Key Result

Theorem 1

Assume that Assumption assumption1 holds true. There exists a unique semigroup $(T_t)_{t\geq 0}$ on $B$ such that, for all $f \in {\cal D}(\mathcal{A}) \coloneqq C_c^{(s)}\cup\{h\}$,

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • Corollary 2
  • Remark 1
  • Remark 2
  • Lemma 1
  • ...and 44 more