The method of time reversal of a Markov process from a cooptional time, introduced by Nagasawa, is used to show that certain occupation time and last exit time problems for one linear diffusion are equivalent to first passage time problems for certain other diffusions. Another proof of Nagasawa's theorem is given, based on the measures of Revuz.
"Some Transformations of Diffusions by Time Reversal." Ann. Probab. 8 (6) 1157 - 1162, December, 1980. https://doi.org/10.1214/aop/1176994576