We present two path decompositions of Markov chains (with general state space) by means of harmonic functions, which are dual to each other. They can be seen as a generalization of Williams’ decomposition of a Brownian motion with drift. The results may be illustrated by a multitude of examples, but we confine ourselves to different types of random walks and the Pólya urn.
"Path decompositions for Markov chains." Ann. Probab. 32 (2) 1370 - 1390, April 2004. https://doi.org/10.1214/009117904000000234