Killing versus catastrophes in birth-death processes and an application to population genetics

We establish connections between the absorption probabilities of a class of birth-death processes with killing, and the stationary tail of a related class of birth-death processes with catastrophes. The major ingredients of the proofs are a decomposition of the dynamics of these processes, a Feynman–Kac type relationship for Markov chains with reset and rebirth, and the concept of Siegmund duality, which allows us to invert the relationship between the processes. We apply our results to a pair of ancestral processes in population genetics, namely the killed ancestral selection graph and the pruned lookdown ancestral selection graph, in a finite population setting and its diffusion limit.

💡 Research Summary

This paper establishes a precise probabilistic correspondence between two families of continuous‑time birth‑death processes: those with a “killing” mechanism (where the entire population can be instantaneously removed) and those with “catastrophes” (where the population can drop by a random amount but never to zero in a single jump). The authors consider a paired setting in which both processes share the same per‑individual birth rates λ_i, death rates μ_i, and a constant catastrophe/killing rate κ, differing only in how the extra transitions are implemented.

The main result (Theorem 1.2) states that, under mild regularity conditions (including divergence of the series ∑1/λ_i and ∑1/μ_i in the infinite‑state case), the stationary tail probabilities a_i of the catastrophe process Z and the absorption probabilities b_i of the killing process X satisfy two explicit algebraic relations: \