  The present paper is devoted to the investigation of the long term behavior
of a class of singular multi-dimensional diffusion processes that get absorbed
in finite time with probability one. Our focus is on the analysis of
quasi-stationary distributions (QSDs), which describe the long term behavior of
the system conditioned on not being absorbed. Under natural Lyapunov
conditions, we construct a QSD and prove the sharp exponential convergence to
this QSD for compactly supported initial distributions. Under stronger Lyapunov
conditions ensuring that the diffusion process comes down from infinity, we
show the uniqueness of a QSD and the exponential convergence to the QSD for all
initial distributions. Our results can be seen as the multi-dimensional
generalization of Cattiaux et al (Ann. Prob. 2009) as well as the complement to
Hening and Nguyen (Ann. Appl. Prob. 2018) which looks at the long term behavior
of multi-dimensional diffusions that can only become extinct asymptotically.
  The centerpiece of our approach concerns a uniformly elliptic operator that
we relate to the generator, or the Fokker-Planck operator, associated to the
diffusion process. This operator only has singular coefficients in its
zeroth-order terms and can be handled more easily than the generator. For this
operator, we establish the discreteness of its spectrum, its principal spectral
theory, the stochastic representation of the semigroup generated by it, and the
global regularity for the associated parabolic equation. We show how our
results can be applied to most ecological models, among which cooperative,
competitive, and predator-prey Lotka-Volterra systems.
