Concentration of quasi-stationary distributions for one-dimensional diffusions with applications

We consider small noise perturbations to an ordinary differential equation (ODE) that have a uniform absorbing state and exhibit transient dynamics in the sense that interesting dynamical behaviors governed by transient states display over finite time intervals and the eventual dynamics is simply controlled by the absorbing state. To capture the transient states, we study the noisevanishing concentration of the so-called quasi-stationary distributions (QSDs) that describe the dynamics before reaching the absorbing state. By establishing concentration estimates based on constructed uniform-in-noises Lyapunov functions, we show that QSDs tend to concentrate on the global attractor of the ODE as noises vanish, and that each limiting measure of QSDs, if exists, must be an invariant measure of the ODE. Overcoming difficulties caused by the degeneracy and singularity of noises at the absorbing state, we further show the tightness of the family of QSDs under additional assumptions motivated by applications, that not only validates a priori information on the concentration of QSDs, but also asserts the reasonability of using QSDs in the mathematical modeling of transient states. Our approaches to the concentration and tightness of QSDs are purely analytic without probabilistic heuristics. Applications to diffusion approximations of chemical reactions and birth-and-death processes of logistic type are also discussed. Rigorously studying the transient dynamics and characterizing the transient states, our study is of both theoretical and practical significance.