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 R d \{0}-valued self-similar Markov processes and S d-1 Rvalued 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.