Siegmund duality with applications to the neutral Moran model conditioned on never being absorbed

We first consider the classical neutral Moran model with two alleles whose fate is either to become extinct or to reach fixation. We study an ergodic version of the Moran model obtained by conditioning it to never hit the boundaries, making use of a Doob transform. We call it the recurrent Moran model. We show that the Siegmund dual of the recurrent Moran process exists and is a substochastic birth and death chain. Conditioning this process to exit in its natural absorbing state, we construct a process with a unique absorbing state which is intertwined to the original recurrent Moran process. The time needed for the intertwined process to first hit its absorbing state is related to the time needed to reach stationarity for the recurrent Moran process. Using spectral information on the intertwined chain, we extract limiting information on this first hitting time that shows that there is no abrupt relaxation to equilibrium for the recurrent Moran chain. This makes use of the relation between duality and intertwining and strong stationary times. Other related transition times of the recurrent Moran chain are also briefly investigated, namely the first return time to the ground state and the expected time needed to move from one end to the other end of the state space.