Quasistationary distributions for one-dimensional diffusions with killing

Details Quasistationary distributions for one-dimensional diffusions with killing Quasistationary distributions for one-dimensional diffusions with killing

2004-06-03

arxiv.org/abs/math/0406052v2

Mathematics

Probability

40 pages, final version accepted for Trans. Amer. Math. Soc. except for a graphic

Scientific paper

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.

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