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

Ellen Baake, Fernando Cordero, Enrico Di Gaspero, Anton Wakolbinger

TL;DR

The paper develops a probabilistic framework linking absorption probabilities of birth-death processes with killing to stationary tail probabilities of paired birth-death processes with catastrophes, via flight representations and Siegmund duality. It introduces a structured decomposition of dynamics and a biased detailed-balance relation for Markov chains with reset and rebirth, enabling a pathwise construction and duality-based inversion between processes. The main result provides explicit factorized relations between $b_i$ and $a_i$, and their inverse, for paired processes, with a rigorous justification using Siegmund duals and flight representations. An application to population genetics demonstrates how the killed ancestral selection graph (kASG) and the pruned lookdown ASG (pLD-ASG) in both finite Moran models and diffusion limits are connected through these dualities, yielding new probabilistic insights into ancestral processes and their equilibrium behavior.

Abstract

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.

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

TL;DR

The paper develops a probabilistic framework linking absorption probabilities of birth-death processes with killing to stationary tail probabilities of paired birth-death processes with catastrophes, via flight representations and Siegmund duality. It introduces a structured decomposition of dynamics and a biased detailed-balance relation for Markov chains with reset and rebirth, enabling a pathwise construction and duality-based inversion between processes. The main result provides explicit factorized relations between and , and their inverse, for paired processes, with a rigorous justification using Siegmund duals and flight representations. An application to population genetics demonstrates how the killed ancestral selection graph (kASG) and the pruned lookdown ASG (pLD-ASG) in both finite Moran models and diffusion limits are connected through these dualities, yielding new probabilistic insights into ancestral processes and their equilibrium behavior.

Abstract

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.
Paper Structure (17 sections, 14 theorems, 101 equations, 9 figures)

This paper contains 17 sections, 14 theorems, 101 equations, 9 figures.

Table of Contents

  1. Introduction and main result
  2. Main result
  3. Outline of the paper.
  4. Proof of Theorem \ref{['thm_general']} A
  5. Markov chains with reset and rebirth, and a biased detailed-balance relation
  6. Flights and Siegmund duality
  7. Flight representations and duality
  8. Flight representation for bdc processes
  9. Siegmund duality between bdc and bdk processes
  10. Proof of Theorem \ref{['thm_general']} B
  11. Siegmund dualities related to $X$ and $Z$
  12. Completing the proof
  13. An application
  14. The two-type Moran model with selection and mutation and its diffusion limit
  15. The connection between kASG and pLD-ASG in the diffusion limit
  16. ...and 2 more sections

Key Result

Theorem 1.2

For $N\in \mathbb N_\infty$, let $X$ and ${Z}$ be a birth-death process with killing and a birth-death process with catastrophes, respectively, paired in the sense of Definition def:paired. In the case $N=\infty$, assume cond_lambda and cond_mu.

Figures (9)

  • Figure 1: The transition graphs of $X^{N}$ (top) and ${Z}^{N-1}$ (bottom) for finite $N$ (see \ref{['X_rates']}, \ref{['calZ_rates']}, and Definition \ref{['def:paired']}).
  • Figure 2: The connections between the paired processes $X$ and $Z$ of Definition \ref{['def:paired']}, the inverse Siegmund dual $X^\circ$ of $X$, and the Siegmund dual ${Z}^\star$ of $Z$; $X^\circ$ is a bdc in the wider sense, as specified in the paragraph below \ref{['calZ_rates']}.
  • Figure 3: The transition graph of ${Z}^{(n)}$ for finite $N$ (see \ref{['qZn_rates']} and proof of Lemma \ref{['projectionlemma']}).
  • Figure 4: The transition graph of $X^{(n)}$ for finite $N$ (see \ref{['qXn_rates']} and proof of Lemma \ref{['hitproblemma']}).
  • Figure 5: The transition graph corresponding to the jump rates $q^\kappa$; arrows representing transitions between states inside the red bubble are omitted.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 20 more