Intrinsic Ultracontractivity for Markov Semigroups and Diffusion Models in Population Dynamics

This paper investigates the long-term behavior of diffusion models in population dynamics before absorption. By establishing the intrinsic ultracontractivity of the associated Markov semigroups, we obtain sharp quantitative estimates for three fundamental limiting objects: the quasi-stationary distribution, the quasi-ergodic distribution, and the mean quasi-ergodic distribution. Specifically, we determine their domains of attraction and prove that convergence to each occurs at an exponential rate. Moreover, we show that the conditioned Q-process is uniformly exponentially ergodic, and we characterize the mixing speed of the original process in terms of the spectral gap. These results provide a unified framework for understanding pre-absorption dynamics in population processes.