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

The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour its solutions. Bertoin Watson (2018) developed a probabilistic approach relying on Feynman-Kac formula, enabled them to answer this for sublinear growth rates. This assumption ensures microscopic remain microscopic. In work, we go further in analysis, assuming bounded fragmentations allowing arbitrarily small reach macroscopic mass finite time. We establish necessary sufficient conditions coefficients ensure Malthusian with exponential speed convergence profile. Furthermore, provide an explicit expression latter.