FRAGMENTATIONS WITH SELF-SIMILAR BRANCHING SPEEDS

We consider fragmentation processes with values in the space of marked partitions of N. i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (Ann. Inst. H. Poincare Prob. Statist. 38, 2002). Our main result is the characterization of these generalized fragmentation processes: a Levy-Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.