On quasi-ergodic distribution for one-dimensional diffusions

In this paper, we study quasi-ergodicity for one-dimensional diffusion $X$ killed at 0, when 0 is an exit boundary and $+\infty$ is an entrance boundary. Using the spectral theory tool, we show that if the killed semigroup is intrinsically ultracontractive, then there exists a unique quasi-ergodic distribution for $X$. An example is given to illustrate the result. Moreover, the ultracontractivity of the killed semigroup is also studied.