Transient one-dimensional diffusions conditioned to converge to a different limit point

@article{Hening2015TransientOD,
  title={Transient one-dimensional diffusions conditioned to converge to a different limit point},
  author={Alexandru Hening},
  journal={arXiv: Probability},
  year={2015}
}
Let $(X_t)_{t\geq 0}$ be a regular one-dimensional diffusion that models a biological population. If one assumes that the population goes extinct in finite time it is natural to study the $Q$-process associated to $(X_t)_{t\geq 0}$. This is the process one gets by conditioning $(X_t)_{t\geq 0}$ to survive into the indefinite future. The motivation for this paper comes from looking at populations that are modeled by diffusions which do not go extinct in finite time but which go `extinct… 
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