Separation of time-scales for the seed bank diffusion and its jump-diffusion limit

We investigate the scaling limit of the seed bank diffusion when reproduction and migration (to and from the seed bank) happen on different time-scales. More precisely, we consider the case when migration is slow' and reproduction is standard’ (in the original time-scale) and then switch to a new, accelerated time-scale, where migration is standard' and reproduction is fast’. This is motivated by models for bacterial dormancy, where periods of quiescence can be orders of magnitude larger than reproductive times, and where it is expected to find non-trivial degenerate genealogies on the evolutionary time-scale. However, the above scaling regime is not only interesting from a biological perspective, but also from a mathematical point of view, since it provides a prototypical example where the expected scaling limit of a continuous diffusion should (and will be) a jump-diffusion. For this situation, standard convergence results often seem to fail in multiple ways. For example, since the set of continuous paths from a closed subset of the c`adl`ag paths in each of the classical Skorohod topologies $J_1,J_2,M_1$ and $M_2$, none of them can be employed for tightness on path-space. Further, a na"ive direct rescaling of the Markov generator corresponding to the continuous diffusion immediately leads to a blow-up of the diffusion coefficient. Still, one can identify a well-defined limit via duality in a surprisingly non-technical way. Indeed, we show that a certain duality relation is in some sense stable under passage to the limit and allows an identification of the limit, avoiding all technicalities related to the blow-up in the classical generator. The result then boils down to a convergence criterion for time-continuous Markov chains in a separation of time-scales regime, which is of independent interest.