Summary: Given an excessive measure \(\eta\) for a right Markov semigroup \((P_ t)\), one can construct a stationary strong Markov process with \(\eta\) as one-dimensional distribution and with \((P_ t)\) as transition semigroup. This process has random birth and death times and the underlying measure space may have infinite mass. The process, whose construction follows easily from a theorem of Kuznetsov, leads to new interpretations of various Riesz type decompositions of the measure \(\eta\) and of the Hunt balayage of \(\eta\) on a set. In addition, it allows one to consider the nontransient case and to provide a last exit type decomposition of \(\eta\).