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

Abstract

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 relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We establish necessary and sufficient conditions on the coefficients of the equation that ensure Malthusian behaviour with exponential speed of convergence to the asymptotic profile. Furthermore, we provide an explicit expression of the latter.

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

“…where we have used the expression of \scrT t given in (29) and the fact that the support of X t \#\delta x0 is the single point \{ X t (x 0 )\} . By Hypothesis 1.3 (in particular since B is continuous on [0, X t (\theta )]), for some C 1 = C 1 (\theta , t) which is increasing in t, we have c(x) = B(x) + \lambda \leq C 1 for all x \in (0, X t (\theta )].…”

“…In [25] a global Doeblin condition is proved (for a similar equation) for kernels \kappa which satisfy, for some \epsilon , \eta , x \ast > 0, the condition that \kappa (x, y) \geq \epsilon for all x \in [0, \eta ] and y \geq x \ast . Here we rather consider kernels that are of self-similar form, which is commonly assumed in the literature about spectral gaps for the growth-fragmentation equation [4,9,15,27,29,61] and includes the classical kernels appearing in applications (in particular equal or unequal mitosis and uniform fragment distribution; see below).…”

“…Regarding non-self-similar kernels, there are results of exponential convergence to the stationary distribution in the literature, but only for bounded fragmentation rates; see [54,62] for PDE-based arguments and [17,19,29] for a probabilistic point of view. We also point out that an optimal condition on the fragment distribution is given in [17] for a spectral gap to exist (for bounded fragmentation rates).…”

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

“…We study the semigroup T by connecting it to that of a Markov process via an h-transform, and this is a feature shared by other recent work such as [10,14,5,13]. However, whereas these previous works have been concerned with finding either a subharmonic function (in the first two cases) or an eigenfunction (in the latter two) for A , we are quite free in our choice of the function h, provided that we verify Assumption 1.…”

“…Other approaches, which do not adapt well to our situation, have been proposed. An approach via Hille-Yosida theory may be found in [4,7,8], and further references therein; a method using strongly continuous semigroups in L 1 spaces is contained in [14,29,27]; [31] discusses perturbation results for C 0 -semigroups in well chosen function spaces; an approach from martingale theory can be found in [10]; and [6] uses a fixed point argument.…”