In this paper we shall derive asymptotic expansions of the Green function and the transition probabilities of Markov additive (MA) processes $(\xi_n, S_n)$ whose first component satisfies Doeblin's condition and the second one takes valued in $Z^d$. The derivation is based on a certain perturbation argument that has been used in previous works in the same context. In our asymptotic expansions, however, not only the principal term but also the second order term are expressed explicitly in terms of a few basic functions that are characteristics of the expansion. The second order term will be important for instance in computation of the harmonic measures of a half space for certain models. We introduce a certain aperiodicity condition, named Condition (AP), that seems a minimal one under which the Fourier analysis can be applied straightforwardly. In the case when Condition (AP) is violated the structure of MA processes will be clarified and it will be shown that in a simple manner the process, if not degenerate, are transformed to another one that satisfies Condition (AP) so that from it we derive either directly or indirectly (depending on purpose) the asymptotic expansions for the original process. It in particular is shown that if the MA processes is irreducible as a Markov process, then the Green function is expanded quite similarly to that of a classical random walk on $Z^d$.
"Asymptotic Estimates Of The Green Functions And Transition Probabilities For Markov Additive Processes." Electron. J. Probab. 12 138 - 180, 2007. https://doi.org/10.1214/EJP.v12-396