Coalescence and sampling distributions for Feller diffusions

Consider the diffusion process defined by the forward equation $u_t(t,x)=\frac{1}{2}\{xu(t,x){\}}_{xx}-\alpha \{xu(t,x){\}}_x$ for $t,x\ge 0$ and $-\mathrm{\infty }<\alpha <\mathrm{\infty }$, 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 $t\to \mathrm{\infty }$. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times.