Scale operators for Cramér--Lundberg type Markov additive processes

In this paper, we consider Markov additive processes (MAPs) whose additive components are $\text{bR}$-valued and have no positive jumps. Specifically, we assume that the modulators are Feller processes with only finitely many jumps on each bounded time interval and that the additive components also have only finitely many jumps on each bounded time interval. For such MAPs, we characterize the associated scale operators and use them to solve the two-sided exit problem, which concerns the first entrance time into one half-line on the event that it is reached before the other, and to characterize the potential measures of the processes killed when their additive components exit an interval. Our proofs rely on properties of analogues of local times, properties of the exit systems obtained by pairing them with suitable kernels, and results on the extension of $C_0$-semigroups to $C_0$-groups.