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The long time behavior of an absorbed Markov process is well described by thelimiting distribution of the process conditioned to not be killed when it isobserved. Our aim is to give an approximation's method of this limit, when theprocess is a 1-dimensional It\^o diffusion whose drift is allowed to explode atthe boundary. In a first step, we show how to restrict the study to the case ofa diffusion with values in a bounded interval and whose drift is bounded. In asecond step, we show an approximation method of the limiting conditionaldistribution of such diffusions, based on a Fleming-Viot type interactingparticle system. We end the paper with two numerical applications : to thelogistic Feller diffusion and to the Wright-Fisher diffusion with values in$]0,1[$ conditioned to be killed at 0.

Approximation of quasi-stationary distributions for 1-dimensional killed diffusions with unbounded drifts