Coalescence and sampling distributions for Feller diffusions

TL;DR

This paper analyzes coalescent processes related to Feller diffusions, providing explicit distributions for ancestral lineages in subcritical and supercritical cases, enhancing understanding of population genetics models.

Contribution

It introduces explicit distributions for ancestral lineages in Feller diffusions, covering both subcritical and supercritical cases, which was not previously detailed.

Findings

Distribution of ancestors at time s for any alpha and initial condition.

Conditional distribution of coalescent trees in subcritical diffusion.

Coalescent times distribution in supercritical diffusion.

Abstract

Consider the diffusion process defined by the forward equation $u_t(t, x) = \tfrac{1}{2}\{x u(t, x)\}_{xx} - \alpha \{x u(t, x)\}_{x}$ for $t, x \ge 0$ and $-\infty < \alpha < \infty$, with an initial condition $u(0, x) = \delta(x - x_0)$. This equation was introduced and solved by Feller to model the growth of a population of independently reproducing individuals. We explore important coalescent processes related to Feller's solution. For any $\alpha$ and $x_0 > 0$ we calculate the distribution of the random variable $A_n(s; t)$, defined as the finite number of ancestors at a time $s$ in the past of a sample of size $n$ taken from the infinite population of a Feller diffusion at a time $t$ since since its initiation. In a subcritical diffusion we find the distribution of population and sample coalescent trees from time $t$ back, conditional on non-extinction as $t \to \infty$. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times.