Asymptotic Analysis of Quasi-limiting Behavior for Drifted Brownian Motion Conditioned to Stay Positive

A Quasi-Stationary Distribution for a Markov process with an almost surely reached absorbing state is a conditionally time-invariant distribution on the state space, which the condition is that the process is not absorbed by the given time. Previous works of Martinez et al. identify the family of Quasi-Stationary Distribution for Brownian motion with negative drift, and characterize the domain of attraction for each of them. This paper will mainly focus on two subjects. 1. We provide a new approach simplifying the existing results, which explains the direct relation between a QSD and an initial distribution in the domain of attraction of the QSD. 2. We will discuss the quasi-limiting behavior of initial distributions that are not in the domain of attraction of any QSD, by finding the right scaling factor and scaling limit of such distributions