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

## TL;DR

This paper studies the exponential functional of Markov Additive Processes, extending existing results, and applies these findings to analyze the genealogy and Hausdorff dimension of multi-type self-similar fragmentation trees.

## Contribution

It extends moment results for exponential functionals of Markov Additive Processes and applies them to multi-type fragmentation trees, including Hausdorff dimension calculations.

## Key findings

- Extended moment formulas for exponential functionals of Markov Additive Processes.
- Derived Hausdorff dimension of genealogical trees in multi-type fragmentation.
- Connected Markov Additive Processes to self-similar fragmentation genealogy.

## Abstract

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 a multi-type L\'evy process: the second component $J$ is a Markov chain on a finite space $\{1,\ldots,K\}$, 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 $I_{\xi}=\int_{0}^{\infty} e^{-\xi_t}\mathrm dt,$ 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.