The strong Malthusian behavior of growth-fragmentation processes

@article{Bertoin2019TheSM,
  title={The strong Malthusian behavior of growth-fragmentation processes},
  author={Jean Bertoin and Alexander R. Watson},
  journal={arXiv: Probability},
  year={2019}
}
Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly; they are used in models of cell division and protein polymerisation. Typically, we may expect that in the long run, the concentrations of cells with given masses increase at some exponential rate, and that, after compensating for this, they arrive at an asymptotic profile. Up to now, this question has mainly been studied for the average behavior of the system, often by means… Expand
Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes.
Growth-fragmentation processes model systems of cells that grow continuously over time and then fragment into smaller pieces. Typically, on average, the number of cells in the system exhibitsExpand
Some Aspects of Growth-Fragmentation
This thesis treats stochastic aspects of fragmentation processes when growth and/or immigration of particles are incorporated as a compensating phenomenon. In a first part, we study the asymptoticExpand
Individual and population approaches for calibrating division rates in population dynamics: Application to the bacterial cell cycle
Modelling, analysing and inferring triggering mechanisms in population reproduction is fundamental in many biological applications. It is also an active and growing research domain in mathematicalExpand
A non-expanding transport distance for some structured equations
Structured equations are a standard modeling tool in mathematical biology. They are integrodifferential equations where the unknown depends on one or several variables, representing the state orExpand
Practical criteria for $R$-positive recurrence of unbounded semigroups
The goal of this note is to show how recent results on the theory of quasi-stationary distributions allow to deduce effortlessly general criteria for the geometric convergence of normalized unboundedExpand
Spectral gap for the growth-fragmentation equation via Harris's Theorem
We study the long-time behaviour of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show theExpand
Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation
We prove, in the framework of measure solutions, that the equal mito-sis equation present persistent asymptotic oscillations. To do so we adopt a duality approach, which is also well suited forExpand

References

SHOWING 1-10 OF 67 REFERENCES
A probabilistic approach to spectral analysis of growth-fragmentation equations
Abstract The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols.Expand
Asymptotics of self-similar growth-fragmentation processes
Markovian growth-fragmentation processes introduced by Bertoin extend the pure fragmentation model by allowing the fragments to grow larger or smaller between dislocation events. What becomes of theExpand
A growth-fragmentation model related to Ornstein–Uhlenbeck type processes
Growth-fragmentation processes describe systems of particles in which each particle may grow larger or smaller, and divide into smaller ones as time proceeds. Unlike previous studies, which haveExpand
Long-Time Asymptotics for Polymerization Models
This study is devoted to the long-term behavior of nucleation, growth and fragmentation equations, modeling the spontaneous formation and kinetics of large polymers in a spatially homogeneous andExpand
Random fragmentation and coagulation processes
Fragmentation and coagulation are two natural phenomena that can be observed in many sciences and at a great variety of scales - from, for example, DNA fragmentation to formation of planets byExpand
Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates
We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving theirExpand
Eigenelements of a General Aggregation-Fragmentation Model
We consider a linear integro-differential equation which arises to describe both aggregation-fragmentation processes and cell division. We prove the existence of a solution (λ, , ϕ) to the relatedExpand
Martingales in self-similar growth-fragmentations and their connections with random planar maps
The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting andExpand
The growth and composition of branching populations
A single-type general branching population develops by individuals reproducing according to i.i.d. point processes on R +, interpreted as the individuals' ages. Such a population can be measured orExpand
On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes
The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and EscobedoExpand
...
1
2
3
4
5
...