Long Time Behavior of General Markov Additive Processes

TL;DR

This paper characterizes the long-term behavior of general Markov additive processes with a Polish space modulator, linking it to ladder times and excursion measures, and applies findings to self-similar Markov processes.

Contribution

It provides a new characterization of the asymptotic behavior of Markov additive processes with Polish space modulator, extending understanding of their long-time dynamics.

Findings

Characterization of long-time behavior via ladder time process and excursion measure.

Application to the asymptotic analysis of self-similar Markov processes.

Validation of assumptions on well-known self-similar processes.

Abstract

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.