Self-similar growth-fragmentation processes with types

In this paper, we are interested in self-similar growth-fragmentation processes with types. It roughly consists of two parts. In the first part, we investigate multitype versions of the self-similar growth-fragmentation processes introduced by Bertoin, therefore extending the signed case of arXiv:2101.02582 to finitely many types. Our main result in this direction describes the law of the spine in the multitype setting. We stress that our arguments only rely on the structure of the underlying Markov additive processes, and hence is more general than arXiv:2101.02582. In the second part, we study $\mathbb{R}^d$-valued self-similar growth-fragmentation processes driven by an isotropic process. These can be seen as multitype growth-fragmentation processes, where the set of types is the sphere $\mathbb{S}^{d-1}$. We give the spinal description in this setting. Finally, we prove that such a family of processes shows up when slicing half-space excursions with hyperplanes.