Quasi-stationarity distributions and diffusion models in population dynamics

In this paper, we study quasi-stationarity for a large class of Kolmogorov diffusions, that is, existence of a quasi-stationary distribution, conditional convergence to such a distribution, construction of a Q-process (process conditioned to be never extinct). The main novelty here is that we allow the drift to go to −∞ at the origin, and the diffusion to have an entrance boundary at +∞. These diffusions arise as images, by a deterministic map, of generalized Feller diffusions, which themselves are obtained as limits of rescaled birth–death processes. Generalized Feller diffusions take non-negative values and are absorbed at zero in finite time with probability 1. A toy example is the logistic Feller diffusion. We give sufficient conditions on the drift near 0 and near +∞ for the existence of quasistationary distributions, as well as rate of convergence, and existence of the Q-process. We also show that under these conditions, there is exactly one conditional limiting distribution (which implies uniqueness of the quasi-stationary distribution) if and only if the process comes down from infinity. Proofs are based on spectral theory. Here the reference measure is the natural symmetric measure for the killed process, and we use in an essential way the Girsanov transform.