The excursion straddling $t$ and the first excursion exceeding $u$ in length of a Markov process are compared in a general setting. These results are then specialized to the excursions from a point, and to specific processes. The main tool is the general theory of excursions as developed by Maisonneuve and others.
"Excursions of a Markov Process." Ann. Probab. 7 (2) 244 - 266, April, 1979. https://doi.org/10.1214/aop/1176995086