Coalescence and sampling distributions for Feller diffusions

Consider the diffusion process defined by the forward equation u_t(t,x)=12{xu(t,x)}_xx-α{xu(t,x)}_x for t,x≥0 and -∞<α<∞, with an initial condition u(0,x)=δ(x-x₀). 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 α and x₀>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 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→∞. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times.