Let $X$ be an $\mu $-symmetric irreducible Markov process on $I$ with strong Feller property. In addition, we assume that $X$ possesses a tightness property. In this paper, we prove some conditional limiting theorems for the process $X$. The emphasis is on conditional ergodic theorem. These results are also discussed in the framework of one-dimensional diffusions.
"Some conditional limiting theorems for symmetric Markov processes with tightness property." Electron. Commun. Probab. 24 1 - 11, 2019. https://doi.org/10.1214/19-ECP265