  In this article, we address the absorption properties of a class of
stochastic differ- ential equations around singular points where both the drift
and diffusion functions vanish. According to the H\"older coefficient alpha of
the diffusion function around the singular point, we identify different
regimes. Stability of the absorbing state, large deviations for the absorption
time, existence of stationary or quasi-stationary distributions are discussed.
In particular, we show that quasi-stationary distributions only exist for alpha
< 3/4, and for alpha in the interval (3/4, 1), no quasi-stationary distribution
is found and numerical simulations tend to show that the process conditioned on
not being absorbed initiates an almost sure exponential convergence towards the
absorbing state (as is demonstrated to be true for alpha = 1). Applications of
these results to stochastic bifurcations are discussed.
