  A Quasi-Stationary Distribution (QSD)for a Markov process with an almost
surely hit absorbing state is a time-invariant initial distribution for the
process conditioned on not being absorbed by any given time. An initial
distribution for the process is in the domain of attraction of some QSD $\nu$
if the distribution of the process a time $t$, conditioned not to be absorbed
by time $t$ converges to $\nu$. In this work study mostly Brownian motion with
constant drift on the half line $[0,\infty)$ absorbed at $0$. Previous work by
Martinez et al. identifies all QSDs and provides a nearly complete
characterization for their domain of attraction. Specifically, it was shown
that if the distribution a well-defined exponential tail (including the case of
lighter than any exponential tail), then it is in the domain of attraction of a
QSD determined by the exponent. In this work we
  1. Obtain a new approach to existing results, explaining the direct relation
between a QSD and an initial distribution in its domain of attraction.
  2. Study the behavior under a wide class of initial distributions whose tail
is heavier than exponential, and obtain no-trivial limits under appropriate
scaling.
