Related papers: On a family of critical growth-fragmentation semig…
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…
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 ($2018$) developed a probabilistic approach…
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…
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…
Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to description of the long time time asymptotics of two critical cases of these equations, when the division…
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. Several important questions about the equation…
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 focused mainly on the self-similar case, we…
This work develops further a probabilist approach to the asymptotic behavior of growth-fragmentation semigroups via the Feynman-Kac formula, which was introduced in a joint article with A.R. Watson [4]. Here, it is first shown that the…
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 exhibits asynchronous exponential growth and, upon…
The subject of this paper is a fragmentation equation with nonconservative solutions, some mass being lost to a dust of zero-mass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the…
We are interested in the large time behavior of the solutions to the growth-fragmentation equation. We work in the space of integrable functions weighted with the principal dual eigenfunction of the growth-fragmentation operator. This space…
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 the known asymptotic behaviors of…
The goal of the present paper is to explore the long-time behavior of the growth-fragmentation equation formulated in the case of equal mitosis and variability in growth rate, under fairly general assumptions on the coefficients. The first…
The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the…
We study the long time behavior of an asymmetric size-structured measure-valued growth-fragmentation branching process that models the dynamics of a population of cells taking into account physiological and morphological asymmetry at…
We study the stochastic growth process in discrete time $x_{i+1} = (1 + \mu_i) x_i$ with growth rate $\mu_i = \rho e^{Z_i - \frac12 var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $dZ_t = - \gamma Z_t dt +…
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 Bertoin, where special attention was…
The asymptotic shape of randomly growing radial clusters is studied. We pose the problem in terms of the dynamics of stochastic partial differential equations. We concentrate on the properties of the realizations of the stochastic growth…
We study the asymptotic behavior of linear evolution equations of the type \partial_t g = Dg + Lg - \lambda g, where L is the fragmentation operator, D is a differential operator, and {\lambda} is the largest eigenvalue of the operator Dg +…
We give here an explicit formula for the following critical case of the growth-fragmentation equation $$\frac{\partial}{\partial t} u(t, x) + \frac{\partial}{\partial x} (gxu(t, x)) + bu(t, x) = b\alpha^2 u(t, \alpha x), \qquad u(0, x) =…