2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72769We 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 Polya urn.Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000234Probability60J10 (Primary) 60J45. (Secondary)text