Abstract
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 L\'evy process with immigration, and apply our result to study the asymptotic properties of the derivative martingale.
Highlights
Fragmentation processes offer a random model for particles which break apart as time passes
One large class of fragmentation models, encompassing the so-called homogeneous fragmentation processes, has been successful, and a comprehensive discussion can be found in the book of Bertoin [6]
Compensated fragmentation processes were defined by Bertoin [7] as a generalisation of homogeneous fragmentations, and permit high-intensity fragmentation and Gaussian fluctuations of the sizes of fragments
Summary
Fragmentation processes offer a random model for particles which break apart as time passes. Our first main result is Theorem 5.2, in which we show that under Pω, the process Z may be regarded as the exponential of a single spectrally negative Lévy process (the spine) with Laplace exponent κ(· + ω) − κ(ω), onto whose jumps are grafted independent copies of Z (under the original measure P). We prove our second main result, Theorem 6.1, which states that the derivative martingale converges to a strictly negative limit under certain conditions This limit is closely related to the process representing the largest fragment of the compensated fragmentation. In the case of compensated fragmentation processes, Dadoun [23] studied the discrete-time skeletons of the derivative martingale via a branching random walk, and used their convergence to obtain asymptotics for the largest fragment.
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