Transient Dynamics of Absorbed Singular Diffusions

We consider a class of one-dimensional absorbed diffusion processes with small singular noises that exhibit multi-scale dynamics in the sense that typical trajectories of the solution process first quickly approach to transient states, captured by quasi-stationary distributions (QSDs), then stay with transient states for a very long period, and finally deviate from transient states and slowly relax to the absorbing state. The main purpose of the present paper is to give qualitative characterizations of such multi-scale dynamics with particular interest in the intriguing transient dynamics governed by transient states. This is achieved by the establishment of (i) noise-vanishing asymptotics of the first eigenvalue and the gap between the first and second eigenvalues of the generator that are respectively the rates for solutions to get absorbed by the absorbing state and attracted by QSDs; (ii) sophisticated estimates quantifying the distance between solutions and convex combinations of QSDs and the absorbing state. Applications to stochastic models arising in chemical reactions and population dynamics are discussed.