Coalescence and sampling distributions for Feller diffusions
Conrad J. Burden, Robert Griffiths · arXiv (Cornell University) · 2022
Consider the diffusion process defined by the forward equation $u_t(t, x) = \tfrac{1}{2}\{x u(t, x)\}_{xx} - α\{x u(t, x)\}_{x}$ for $t, x \ge 0$ and $-\infty 0$ 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.