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A Markov Additive Process is a bi-variate Markov process$(\xi,J)=\big((\xi_t,J_t),t\geq0\big)$ which should be thought of as amulti-type L\'evy process: the second component $J$ is a Markov chain on afinite space $\{1,\ldots,K\}$, and the first component $\xi$ behaves locally asa L\'evy process, with local dynamics depending on $J$. In thesubordinator-like case where $\xi$ is nondecreasing, we establish severalresults concerning the moments of $\xi$ and of its exponential functional$I_{\xi}=\int_{0}^{\infty} e^{-\xi_t}\mathrm dt,$ extending the work of Carmonaet al., and Bertoin and Yor.  We then apply these results to the study of multi-type self-similarfragmentation processes: these are self-similar analogues of Bertoin'shomogeneous multi-type fragmentation processes Notably, we encode the genealogyof the process in a tree, and under some Malthusian hypotheses, compute itsHausdorff dimension in a generalisation of our previous work.

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