Some applications of quasi-boundedness for excessive measures

Abstract

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

The research of both authors was supported, in part, by NSF Grant DMS 91-01675.