This paper provides an explanation of Siegmund's duality for absorbing and reflecting Markov processes by means of a graphical representation of the type used in the analysis of infinite particle systems. It is shown that coupled realisations of a Markov process conditioned to start at each of the points of the state space can be generated on the same probability space in such a way that their ordering is preserved. Using the same probability space a specific construction is then given for the dual process.
"A Sample Path Proof of the Duality for Stochastically Monotone Markov Processes." Ann. Probab. 13 (2) 558 - 565, May, 1985. https://doi.org/10.1214/aop/1176993008