Path decompositions for Markov processes

Paths decompositions of Markov processes have been studied in various instances, beginning with David Williams' decomposition of Brownian motion with drift. We prove a general result for strong Markov processes $X$ possessing a positive harmonic function $h$. The paths of $X$ are split into the parts before and after the moment, when $h(X)$ takes its maximum for the first time. The results are explained for continuous processes as well as general Markov processes and exemplified by a general version of Williams' result, killed Brownian motion, a last exit decomposition for Brownian motion and a decomposition for space-time Brownian bridge.