Random walk Green kernels in the neutral Moran model conditioned on survivors at a random time to origin

In the theory of finite discrete-time birth and death chains with absorbing endpoint boundaries, the evaluation of both additive and multiplicative path functionals is made possible by their Green and -potential kernels. These computations are addressed in the context of such Markov chains. The application to the neutral Moran model of population genetics yields first hitting and return times. A neutral Moran bridge model, forward and backward in time, for a given total number x of survivors of a single common ancestor at some random time T to the origin of times, yields the age of a mutant allele currently observed to have x copies of itself. This forward theory of age, made possible by Green kernels, is comparable to Watterson's backward theory of age, which makes use of the reversibility of the Moran chain.