Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Feller diffusion as the limit of a coalescent point process

Conrad J. Burden, Robert C. Griffiths

TL;DR

This paper treats the Feller diffusion as the diffusion limit of a coalescent point process (CPP) in which the node-height density concentrates toward zero, producing trees that come down from infinity. By embedding Bernoulli sampling into this diffusion limit via Poisson sampling and leveraging Wiuf's node-height parameterization, it provides unified, explicit forms for coalescent-time distributions under various sampling schemes and conditioning. It recovers and extends known results for near-critical and large-population birth-death processes, including joint densities conditioned on current population size and $k$-sampled trees, with efficient iterative and hypergeometric expressions for expected waiting times. The framework highlights symmetries between super- and sub-critical regimes and yields practical methods to derive coalescent properties across BD and diffusion models, enabling broader applicability to diffusion-branching processes in population genetics. Its results offer a cohesive bridge between discrete branching-process limits and diffusion-level coalescent descriptions, with potential for further extensions to diffusion-conditioned sampling scenarios.

Abstract

The Feller diffusion is studied as the limit of a coalescent point process in which the density of the node height distribution is skewed towards zero. Using a unified approach, a number of recent results pertaining to scaling limits of branching processes are reinterpreted as properties of the Feller diffusion arising from this limit. The notion of Bernoulli sampling of a finite population is extended to the diffusion limit to cover finite Poisson-distributed samples drawn from infinite continuum populations. We show that the coalescent tree of a Poisson-sampled Feller diffusion corresponds to a coalescent point process with a node height distribution taking the same algebraic form as that of a Bernoulli-sampled birth-death process. By adapting methods for analysing k-sampled birth-death processes, in which the sample size is pre-specified, we develop methods for studying the coalescent properties of the k-sampled Feller diffusion.

The Feller diffusion as the limit of a coalescent point process

TL;DR

This paper treats the Feller diffusion as the diffusion limit of a coalescent point process (CPP) in which the node-height density concentrates toward zero, producing trees that come down from infinity. By embedding Bernoulli sampling into this diffusion limit via Poisson sampling and leveraging Wiuf's node-height parameterization, it provides unified, explicit forms for coalescent-time distributions under various sampling schemes and conditioning. It recovers and extends known results for near-critical and large-population birth-death processes, including joint densities conditioned on current population size and -sampled trees, with efficient iterative and hypergeometric expressions for expected waiting times. The framework highlights symmetries between super- and sub-critical regimes and yields practical methods to derive coalescent properties across BD and diffusion models, enabling broader applicability to diffusion-branching processes in population genetics. Its results offer a cohesive bridge between discrete branching-process limits and diffusion-level coalescent descriptions, with potential for further extensions to diffusion-conditioned sampling scenarios.

Abstract

The Feller diffusion is studied as the limit of a coalescent point process in which the density of the node height distribution is skewed towards zero. Using a unified approach, a number of recent results pertaining to scaling limits of branching processes are reinterpreted as properties of the Feller diffusion arising from this limit. The notion of Bernoulli sampling of a finite population is extended to the diffusion limit to cover finite Poisson-distributed samples drawn from infinite continuum populations. We show that the coalescent tree of a Poisson-sampled Feller diffusion corresponds to a coalescent point process with a node height distribution taking the same algebraic form as that of a Bernoulli-sampled birth-death process. By adapting methods for analysing k-sampled birth-death processes, in which the sample size is pre-specified, we develop methods for studying the coalescent properties of the k-sampled Feller diffusion.
Paper Structure (18 sections, 7 theorems, 106 equations, 3 figures)

This paper contains 18 sections, 7 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Coalescent point processes
  3. Coalescent time distribution given $T_1 = t$
  4. Coalescent time distribution with uniform prior on $T_1$
  5. Iterative formulae for expected coalescent times and waiting times for a general CPP
  6. Birth-death process as a CPP
  7. Feller diffusion as the limit of a CPP
  8. Sampling
  9. Bernoulli sampling for a BD process
  10. Poisson sampling for a Feller diffusion
  11. $k$-sampling for a Feller diffusion
  12. Distribution of coalescent times for a $k$-sampled Feller diffusion without conditioning on $X(t) = x$
  13. Distribution of coalescent times for a $k$-sampled Feller diffusion with conditioning on $X(t) = x$
  14. Expected inter-coalescent waiting times for trees constructed from Wuif's node-height distribution
  15. Conclusions
  16. ...and 3 more sections

Key Result

Lemma 1

The reconstructed tree of a BD process $(N(u))_{u \in [0, t]}$ with constant birth rate $\lambda$ and death rate $\mu$ stopped at time $t$ corresponds to a CPP with inverse tail distribution where $\alpha = \lambda - \mu$. The corresponding node height cdf is

Figures (3)

  • Figure 1: A CPP with $n = 6$ and the associated ultrametric tree.
  • Figure 2: (a) The coalescent tree of a Bernoulli-sampled BD process or Poisson-sampled Feller diffusion conditioned on a sample of size 20 for (a) $\gamma = 1000$, (b) $\gamma = 1$ (equivalent to a rate-1 Yule process), and (c) $\gamma = 0.05$.
  • Figure 3: Section of a population coalescent tree in black and the embedded sample coalescent tree in red. In this example the sample coalescent times $\tilde{T}_k$ and $\tilde{T}_{k - 1}$ are shown for $k = 3$. We have $m_{k - 1} = 4, m_k = 7$, and hence $\tilde{T}_3 = T_5$ and $\tilde{T}_4 = T_8$.

Theorems & Definitions (14)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Corollary 1
  • Theorem 3
  • ...and 4 more