Multitype self-similar growth-fragmentation processes

In this paper, we are interested in multitype self -similar growth -fragmentation processes. More precisely, we investigate a multitype version of the self -similar growth -fragmentation processes introduced by Bertoin, where the type of the particles may also evolve in time according to a Markov chain. This extends the signed case (considered by the first author in a previous work) to the case of finitely many types. Our main result in this direction describes the law of the spine in the multitype setting. In order to do so, we introduce two genealogical martingales, in the same spirit as in the positive case, which allow us not only to obtain the law of the spine but also to study the limit of the empirical measure of fragments. We stress that our arguments only rely on the structure of the underlying Markov additive processes (MAPs), and hence is more general than the treatment of the signed case. Our methods also require new results on exponential functionals for MAPs and a multitype version of the tail estimates in multiplicative cascades which are interesting in their own right.