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

TL;DR

This paper develops a general many-to-few formula for non-local branching Markov processes, enabling the computation of expectations over multiple population states and analyzing the long-term behavior of the process.

Contribution

It extends the many-to-few formula to non-local branching Markov processes and derives asymptotic laws for the population and common ancestor times.

Findings

Derived a general formula for the limiting law of the death time of the most recent common ancestor.

Established the asymptotic distribution of population sizes at two different times.

Extended previous formulas to non-local branching processes.

Abstract

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 [14] 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 tends to infinity. Moreover, we describe the limiting law of the population sizes at two different times, in the same asymptotic regime.