On the exponential functional of Markov Additive Processes, and applications to multi-type self-similar fragmentation processes and trees

A Markov Additive Process is a bi-variate Markov process $(\xi ,J)=(({\xi }_t,J_t),t\ge 0)$ which should be thought of as a multi-type L'evy process: the second component $J$ is a Markov chain on a finite space $\{1,\dots ,K\}$, and the first component $\xi$ behaves locally as a L'evy process, with local dynamics depending on $J$. In the subordinator-like case where $\xi$ is nondecreasing, we establish several results concerning the moments of $\xi$ and of its exponential functional $I_{\xi }={\int }_0^{\mathrm{\infty }}{e}^{-{\xi }_t}\mathrm{d}t,$ extending the work of Carmona et al., and Bertoin and Yor. We then apply these results to the study of multi-type self-similar fragmentation processes: these are self-similar analogues of Bertoin’s homogeneous multi-type fragmentation processes Notably, we encode the genealogy of the process in a tree, and under some Malthusian hypotheses, compute its Hausdorff dimension in a generalisation of our previous work.