Singularities of the generator of a Markov additive process with one-sided jumps

We analyze the number of zeros of det(F(alpha)), where F(alpha) is the matrix cumulant generating function of a Markov Additive Process (MAP) with one-sided jumps. The focus is on the number of zeros in the right half of the comple x plane, where det(F(alpha)) is well-defined. Moreover, we analyze the case of a killed MAP with state-dependent killing rates, and the limiting behavior of the zeros as all killing rates converge to 0. We argue that our results are particulary useful for the fluctuation theory of MAPs. For example, they lead, under mild assumptions, to a straightforward identification of the stationary distribution of a reflected MAP with one-sided jumps.