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Multitype $Λ$-coalescents and continuous state branching processes

Adrián González Casanova, Noemi Kurt, Imanol Nuñez Morales, José Luis Pérez

TL;DR

This work extends the probabilistic links between coalescent theory and continuous-state branching processes to the multitype setting. By coupling two independent multitype CSBPs through a sequential sampling (culling) procedure, the authors construct an autonomous multitype frequency process whose moment dual is the block-counting process of a multitype $\Lambda$-coalescent; they then demonstrate a precise homeomorphism between the parameter spaces of multitype CSBPs and multitype $\Lambda$-coalescents via an explicit transformation (Gillespie-type). This yields both forward-in-time population dynamics and backward-in-time genealogical structures, with convergence results and a robust topological framework that generalizes previous one-dimensional results. The paper also clarifies when the Gillespie multitype coalescents align with the Johnston–Kyprianou–Rogers class and points to natural open questions about Pitman-sense generalisations and the broader mapping of CSBPs to multityype coalescents. Overall, the results deepen probabilistic connections between multitype branching mechanisms and multitype coalescent continuity, providing tools for studying complex structured populations.

Abstract

We provide new connections between multitype $Λ$-coalescents and multitype continuous state branching processes via duality and a homeomorphism on their parameter space. The approach is based on a sequential sampling procedure for the frequency process of independent CSBPs, and provides forward and backward processes for multitype population models under $Λ$-type reproduction. It provides some insight on different approaches to generalise $Λ$-coalescents to the multitype setup.

Multitype $Λ$-coalescents and continuous state branching processes

TL;DR

This work extends the probabilistic links between coalescent theory and continuous-state branching processes to the multitype setting. By coupling two independent multitype CSBPs through a sequential sampling (culling) procedure, the authors construct an autonomous multitype frequency process whose moment dual is the block-counting process of a multitype -coalescent; they then demonstrate a precise homeomorphism between the parameter spaces of multitype CSBPs and multitype -coalescents via an explicit transformation (Gillespie-type). This yields both forward-in-time population dynamics and backward-in-time genealogical structures, with convergence results and a robust topological framework that generalizes previous one-dimensional results. The paper also clarifies when the Gillespie multitype coalescents align with the Johnston–Kyprianou–Rogers class and points to natural open questions about Pitman-sense generalisations and the broader mapping of CSBPs to multityype coalescents. Overall, the results deepen probabilistic connections between multitype branching mechanisms and multitype coalescent continuity, providing tools for studying complex structured populations.

Abstract

We provide new connections between multitype -coalescents and multitype continuous state branching processes via duality and a homeomorphism on their parameter space. The approach is based on a sequential sampling procedure for the frequency process of independent CSBPs, and provides forward and backward processes for multitype population models under -type reproduction. It provides some insight on different approaches to generalise -coalescents to the multitype setup.
Paper Structure (15 sections, 11 theorems, 121 equations)

This paper contains 15 sections, 11 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Multitype L-coalescents
  4. Continuous state branching processes
  5. Multitype population model and the sequential sampling procedure
  6. Main results: Duality and homeomorphism
  7. Proof and discussion of the homeomorphism
  8. Homeomorphism between parameter spaces of CSBPs and multitype L-coalescents
  9. The topological space of multitype L-coalescents
  10. The topological space of multitype CSBPs
  11. Frequency process and convergence the sequential sampling procedure
  12. Pathwise uniqueness
  13. Existence of a strong solution
  14. Sequential sampling
  15. Proof of Theorem \ref{['theo:block_duality']}

Key Result

Theorem 2.1

For any $z \in (0, \infty)^d$ and $r \in [0, 1]^d$, there exists a unique strong solution $R^{(z,r)}$ to eq:sde_d_colonies with initial condition $R^{(z,r)}(0) = r$. Additionally, $R^{(z,r)}$ is a Feller process, and given $z \in (0,\infty)^d$ and $T > 0$, $\overline{R}^{n} \Rightarrow R^{(z,r)}$ as

Theorems & Definitions (21)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.2
  • ...and 11 more