Many-to-few for non-local branching Markov process

We provide a many -to -few formula in the general setting of non -local branching Markov processes. This formula allows one to compute expectations of k -fold sums over functions of the population at k different times. The result generalises [13] to the non -local setting, as introduced in [11] and [8]. As an application, we consider the case when the branching process is critical, and conditioned to survive for a large time. In this setting, we prove a general formula for the limiting law of the death time of the most recent common ancestor of two particles selected uniformly from the population at two different times, as t -> infinity. Moreover, we describe the limiting law of the population sizes at two different times, in the same asymptotic regime.