# Probabilistic aspects of critical growth-fragmentation equations

@article{Bertoin2016ProbabilisticAO, title={Probabilistic aspects of critical growth-fragmentation equations}, author={Jean Bertoin and Alexander R. Watson}, journal={Advances in Applied Probability}, year={2016}, volume={48}, pages={37 - 61} }

Abstract The self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered in Doumic and Escobedo (2015) for the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a…

## 29 Citations

Local explosion in self-similar growth-fragmentation processes

- Mathematics
- 2016

Markovian growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. They were introduced in a work of…

Some Aspects of Growth-Fragmentation

- Mathematics
- 2019

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 asymptotic…

A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates

- MathematicsESAIM: Probability and Statistics
- 2021

The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson…

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

- Mathematics
- 2022

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…

Martingales in self-similar growth-fragmentations and their connections with random planar maps

- Mathematics
- 2016

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 and…

A growth-fragmentation model related to Ornstein–Uhlenbeck type processes

- Mathematics
- 2017

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 have…

Spectral Gap for the Growth-Fragmentation Equation via Harris's Theorem

- MathematicsSIAM Journal on Mathematical Analysis
- 2021

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 the…

Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat

- Mathematics
- 2022

We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from…

Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations

- MathematicsArXiv
- 2015

Using the Mellin transform of the equation, the long time behavior of the solutions of the fragmentation and growth-fragmentation equations is determined and the results show the strong dependence of this asymptotic behavior with respect to the initial data.

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