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Probability tilting of compensated fragmentations. (English) Zbl 1466.60097

Summary: Fragmentation processes are part of a broad class of models describing the evolution of a system of particles which split apart at random. These models are widely used in biology, materials science and nuclear physics, and their asymptotic behaviour at large times is interesting both mathematically and practically. The spine decomposition is a key tool in its study. In this work, we consider 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 Lévy process with immigration, and apply our result to study the asymptotic properties of the derivative martingale.

MSC:

60G51 Processes with independent increments; Lévy processes
60J25 Continuous-time Markov processes on general state spaces
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

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