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Asymptotic conditional distributions related to one-dimensional generalized diffusion processes. (English) Zbl 1122.60068
Let $$L:=(d/dm)(d/ds)$$ be a generalized diffusion operator on a fixed open interval, and $$\{X(t): t\geq 0, \mathbb{P}_x: x\in I_m\}$$ be a generalized diffusion process with the generator $$L$$, where $$I_m$$ is the support of the speed measure $$dm$$. The authors investigate the asymptotic behavior of the conditional expectations $\mathbb{E}_x(f(X(\tau t))\mid t<\sigma_{\inf I_m}\wedge \sigma_{\sup I_m}) \quad \text{and} \quad \mathbb{E}_x(f(X(\tau t))\mid t<\sigma_{\inf I_m}< \sigma_{\sup I_m}),$ where $$\sigma_a$$ is the first hitting time of $$a\in I_m$$. The paper closes with an application of the main results to population genetics models.

##### MSC:
 60J60 Diffusion processes 60F05 Central limit and other weak theorems 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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