## Asymptotic estimates of the Green functions and transition probabilities for Markov additive processes.(English)Zbl 1134.60055

The author studies a Markov additive process whose first component satisfies Doeblin’s condition and the second one takes values in the $$d$$-dimensional lattice $$\mathbb Z^d$$. By using a certain perturbation technique, the author obtained asymptotic expansions of the Green function and the transition probabilities of the Markov additive process. In the asymptotic expansion, the first and the second order terms are expressed in terms of a few basic functions that are characteristics of the expansion.

### MSC:

 60K15 Markov renewal processes, semi-Markov processes 60J45 Probabilistic potential theory 60J05 Discrete-time Markov processes on general state spaces
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