On a family of critical growth-fragmentation semigroups and refracted L\'evy 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 Escobedo ($2016$) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely \begin{equation} c(x) = \begin{cases} a_{_-} x \quad \quad x < 1 \\ a_{_+} x \quad \quad x \geq 1, \end{cases} \end{equation} with $0 < a_{_+} < a_{_-}$. We give necessary and sufficient conditions on the coefficients ensuring the Malthusian behaviour with exponential speed of convergence to an asymptotic profile, and also provide an explicit expression of the latter. Our approach relies crucially on properties of so-called refracted L\'evy processes that arise naturally in this setting.