New approach to spectral theory of generalized second-order differential operators and its applications to probability theory

众所周知，线性扩散用广义二阶微分算子来描述。因此，对各种量的线性扩散的研究归结为关于谱函数的问题。在本研究中，我们得到了谱函数渐近的一个充要条件。此外，我们还得到了关于扩散的最大过程的一些相关结果。

It is well known that a linear diffusion is described by a generalized second-order differential operator. Therefore, the study of various quantities of linear diffusions is reduced to problems on the spectral functions. In the present study we obtained a necessary and sufficient condition on the asymptotic behavior of the spectral function. Furthermore, we also obtained some related results on the maximum process of the diffusion.