  We extend some results on the convergence of one-dimensional diffusions
killed at the boundary, conditioned on extended survival, to the case of
general killing on the interior. We show, under fairly general conditions, that
a diffusion conditioned on long survival either runs off to infinity almost
surely, or almost surely converges to a quasistationary distribution given by
the lowest eigenfunction of the generator. In the absence of internal killing,
only a sufficiently strong inward drift can keep the process close to the
origin, to allow convergence in distribution. An alternative, that arises when
general killing is allowed, is that the conditioned process is held near the
origin by a high rate of killing near infinity. We also extend, to the case of
general killing, the standard result on convergence to a quasistationary
distribution of a diffusion on a compact interval.
