Probability tilting of compensated fragmentations

Fragmentation processes are part of a broad class of models describing the evolution of a system of particles which split apart at random. Their asymptotic behaviour at large times is of crucial interest, and the spine decomposition is a key tool in its study. In this work, we study the class of compensated fragmentations, or homogeneous growth-fragmentations, recently defined by Bertoin. We give a complete spine decomposition of these processes in terms of a L\'evy process with immigration, and apply our result to study the asymptotic properties of the derivative martingale.