Long Time Behavior of General Markov Additive Processes

We study general Markov additive processes when the state space of the modulator is a Polish space. Under some regularity assumptions, our main result is the characterization of the long-time behavior of the ordinate in terms of the associated ladder time process and the excursion measure. An important application of Markov additive processes is the Lamperti-Kiu transform, which gives a correspondence between $\mathbb{R}^d\backslash \{0\}$-valued self-similar Markov processes and $S^{d-1}\times \mathbb{R}$-valued Markov additive processes. The asymptotic behavior of the radial distance from the origin of a self-similar Markov process can be characterized by the long-time behavior of the ordinate of the corresponding Markov additive process. We show the applicability of our assumptions on some well-known self-similar Markov processes.