Non regular limits of one-dimensional generalized diffusion processes

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We considered the phenomena where we have non regular limits of one-dimensional generalized diffusion processes. When the limit of scale functions and that of speed measure functions have common discontinuous points, we proved the convergence of transition probabilities and showed that the limit process can be a one-dimensional bi-generalized diffusion process, by means of convergence of Green functions. We also investigated the effect of boundary conditions on harmonic transform of one-dimensional generalized diffusion processes. Further we showed some limit theorems for Moran model in population genetics.