  The quantitative long time behavior of absorbing, finite, irreducible Markov
processes is considered. Via Doob transforms, it is shown that only the
knowledge of the ratio of the values of the underlying first Dirichlet
eigenvector is necessary to come back to the well-investigated situation of the
convergence to equilibrium of ergodic finite Markov processes. This leads to
explicit estimates on the convergence to quasi-stationarity, in particular via
functional inequalities. When the process is reversible, the optimal
exponential rate consisting of the spectral gap between the two first Dirichlet
eigenvalues is recovered. Several simple examples are provided to illustrate
the bounds obtained.
