A simple condition sufficient to ensure that a function of a time-homogeneous Markov process is again a time-homogeneous Markov process is proved. This result is then used to study a number of diffusions; in particular, an extension of a result of Pitman is proved, from which it is possible easily to deduce the path decompositions of Williams.
"Markov Functions." Ann. Probab. 9 (4) 573 - 582, August, 1981. https://doi.org/10.1214/aop/1176994363