Some applications of quasi-boundedness for excessive measures

Let ~ and m be excessive measures for a right Markov process X and let Q~ and Qm be the associated stationary Kuznetsov processes. We show that if ~ and m are harmonic, then Q~ ~ Qm if and only if ~ is quasi-bounded by m in the sense that ~ = where each term in the sum is an excessive measure dominated by m. This result allows us to describe the Lebesgue decomposition of Q~ relative to Qm and to give an explicit formula for the Radon-Nikodym derivative in case Q~ « Qm. As a second application of quasi-boundedness for excessive measures, we obtain a general form of a theorem of U. Kuran, in which regularity for the Dirichlet problem is characterized by the quasi-boundedness of a suitable excessive measure.